GiorgaKiSs - you are a pleasure! (That means I like you.)
Apologies for my expression - and please continue to ask. Whilst American movie stars think foreigners understand English if they just speak s-l-o-w-l-y. I'm Australian and now that nobody understands us (eh mate?) even though I live in the second biggest Greek city in the world (or was).
(And I won't post any smart Greek words. Last time I tried that in Swedish I got gay confused with horny. (Isn't gay related to the Mallacan Straits?)
By that ground's "accidental inclusion" I merely meant that diagrammatically, it shows the red +12V connected to black 0V (ground).
But I assumed that was a error, else meant pehaps you were using shielded cable etc.
But you are correct - the screen must be grounded and from its power supply .
If it was not grounded, it would find some other grounding path and probably blow its components. That's bad for most input and output circuits.
(Though by lucky coincidence, VGA and similar interfaces often use a shielded cable. The outer shield connects each end's VGA-connector's ground together (and these are often chassis ground) and thus can ground the screen. (The shield being relatively heavy and carry more current than other signals.) Other cables can do the same - eg Ethernet etc Cat-5 cables if all "returns" for each signal wire are connected together and to ground.)
BTW - this relates to the separate termination of signal grounds and power grounds even though somewhere they will be connected together (but only from one interconnection) - by that's another story!
By commoned together, I simply mean connected together.
EG - your amp supplies should be common = connected together - ie, using that power distributor. It seems "we" also think the PSU (Power Supply Unit - ie (1) Car PC Pwr Supply) should be separate to the amps - or don't need the cap(acitor). Hence common = connect the PSU's power input to the screen's power input - as if they had their own Distribution Block (power distributor).
That makes sense from a general electrical and system distribution POV (Point Of View).
IE - the audio system is it's own distribution system and separate from the PC's components and system.
Each would probably have their own (separate) fuse(s) so that if the audio overloads, it merely blows its fuse, but not the PC's fuse.
(Wouldn't you hate it if a fault in the smaller or less important amplifier blew the fuse that also protected the PC? The PC dies just as Greece kicks the winning goal against the Turks!! Or better still - the Swedes! Yay!)
So....what for the "? fuse" 1A or 5~10 A ????
Aha! Basic fusing (aka protection) theory! (aka = also known as; aka "(is) also called")
1 - Equipment Protection - each piece of equipment should have its own protection - ie "blow its own fuse" - rather than destroy itself.
Most overload sensitive or fault-prone devices have their own protection. EG - amplifiers, televisions, house electrics. Their fuses may be user-changeable and easy to get to (amps, house) but may be internal when they are not expected to blow, but are still the sacrificial component that is sacrificed to the Great Smoke God (eg, multimeters aka DMMs, televisions, radios).
If an equipment does not have its own fuse, then it may be grouped with others. This is seen in car electrics - many circuits might share the same fuse. The fuse is big enough to handle all the different loads (circuits), but small enough so that if a fault on any circuit - like a short to ground - will blow the fuse. (Note that when shared like this, circuits are usually "grouped" into related functional groups. Or sometimes deliberately separated - eg left headlights from right, or hi-beam form low-beam - so that if one blows, you still have something available or some alternative.)
If equipment does NOT have its own protection, you may have to provide it.
2 - Distribution Protection - the protection of the power sources, cables and wires.
If you short a car battery, you will have massive sparks, heat, and maybe an explosion. Hence you must have protection that avoids that.
In old days, the protection was the "physical security" of the distribution (wires) from the battery to the fuses (often in the cabin).
These days, there are usually main or master fuses - often call fuselinks - at or near the battery to provide electrical protection (aka security) from battery shorts etc.
But those fuselinks (aka flinks) are also sized to protect the "downstream" wires that go to the amps or Distribution Blocks or fuse blocks etc.
EG - a 1000W RMS amplifier may have a 150A cable supplying it from the battery, so there might be a 150A (or smaller) flink or fuse or circuit-breaker at the start of the 150A cable.
That 150A cable might be supplying your amps instead. The 150A fuse is NOT to protect your amps, but the distribution to them - your amps have their own protection as per (1) above. [If your amps are only 40A, you might use a 40A or 50A or 60A etc for the 150A cable. The 150A cable is still good because it means less resistance than a 40A cable, but you may niot need and expensive 150A fuse. Using a 40A fuse "upstream" may save using a 40A downstream fuse at the amp end.]
Power distribution is often a "fan-out" situation - eg, an 80A main circuit splits into 2x40A circuits and one of those 40A branches into 3x10A & 20A & 5A etc.
Each sized cable should be protected upstream. IE - the 5A circuit with 5A cable has its 5A fuse which is powered from the 40A cable & its fuse fro the 80A cable and fuse.
Power distribution can be tricky because you want the downstream fuses (eg 5A) to blow before the upstream fuses (40A & 80A). (This may seem ridiculous, but because fuses have different response characteristics - it can happen. It's probably a more common problems with (mismatched) circuit breakers.)
The upstream fuse may be smaller than the total of the downstreams because provided it protects its immediate cable, it may be setting some maximum current for all the downstream loads (ie, no individual fault, but the lot is taking too much power/current).
Likewise the upstream fuse may be larger than the downstreams because it is merely protecting its own wire.
And as per the amp example earlier, of the largest or only downstream fuse is 5A, then the upstream 80A fuse could be replaced by a 5A fuse and the downstream 5A fuse omitted. This is common in "minimal resistance" distributions such as audio systems whereby you remove one fuse (resistance) from the chain. (Not that you would do that for a 5A circuit, but I'm merely explaining the principal.)
Now all the above crap is simply summarised as "each fuse protects its downstream load" where load includes wire/cables and shorts, downstream means "until the next fuse", and fuse means protection - circuit breakers, flinks fuses etc.
But I wanted to detail different scenarios because (1) I have a rambling reputation, (2) I like to see people relax and catch up on well earned sleep, (3) I see some weird info around the place and want to overcome that, and teach people etc (else generate discussion etc).
Another aspect is protection ratings - a general ROT (Rule Of Thumb) is that protection should normally be operating on average at not more than 70% of its rating.
IE - a 10A fuse for a 7A average load (that can peak at 9A, and probably 12A for a few minutes depending on the fuse characteristic); but a 9A average load should be using a 15A fuse, not 10A.
But some circuits can be more critical than that, but the they probably would not use fuses.
Keep in mind that a fuse won't blow at a 101% overload, and it may take an hour at 110%, 30 minutes at 120% and 5 secs at 150% overload etc.
Then there are (normal) fast blow fuses, slo-blo fuses (for motors and hi in-rush currents), etc.
So, in answer to So....what for the "? fuse" 1A or 5~10 A ????, if the screen has its own fuse (and I expect it would - and hopefully externally accessible unless it is unlikely to blow), then you only protect the wire to it (maybe 5A etc?).
If no fuse - or if an internal fuse - you may want to use a 1A fuse assuming that's the max current for the screen (or that 0.7A is).
Its 9W rating is probably at 13.8V. That means 0.65A (P-VI, hence I=P/V = 9W/13,8V = .65A)
If a "constant power" load and it can run on 9V, then it 9W = 9W/9V - 1A.
But is that 9W rating its average or maximum consumption? Does it not have any in-rush current?
Hence why it's nice to be able to easily change fuses - at least until stiff like hat gets sorted out.
FYI - that M4 PSU mentioned operating down to 6V input.
Assume it is 250W input at 6V (probably providing 225W if it is 90% efficient), that is 250W/6V = 42Amps (say a 40A fuse) as opposed to 250/12 & 250/14 - 21A & 17A (a 25A fuse) at 12V and 14V respectively. (12V being maybe typical supply voltage, engine off, & 14V engine on.)
So the PSU (for 225W output assuming 90% efficiency = 250W input) - a 25A fuse, or 40A fuse?
Answer??? The PSU should have its own protection, hence wire & it fuse or for worst case - lets make it 50A.
But it is likely 25A or 30A is enough since 6V means a real bad battery under heavy cranking, and normally cranking wouldn't be for long and rarely below 10 volts, maybe 8V. And since fuses handle overloads for short periods....
And if you use a circuit breaker for the PSU and PC - I'd advise a manual reset. You don't want an off-on-off-on-off situation for a PC!
Welcome to the fun and simple world of system power wiring. (Where planning saves much $moke & dollar$, and you find out that a wire and a load is merely thing that contains smoke which is dying to be sacrificed and released to the Great Smoke God that resides in our wallets and heaven.)
I'll let others suggest wiring to battery though I reckon yes - the battery is the "cleanest" power source (a battery is a big cap that filters most noise).
But there are also circuits that connect power the PSU after charging begins (that latch on, and have manual initiation/bypass too) or you may want a second battery as is often used for hi-power audio, or independence from the main battery (in case the PC etc flattens the battery).
And there are low-voltage cutout switches (for ~$20) that can control relays....
But plan (which as SoundMan said - you did very well! Nice to see!), iterate (ie, change and improve), review, re-iterate, review, lose temper, argue.... get concensus/approval or make a decision ('cos we'll be - or rather, I'll be bickering forever) and get the ultimate design.
Then you can make the ultimate sacrifice to Smokey! (aka "All care & no responsibility" aka "we don't care and it's your responsibility anyhow").