In this video:
https://www.lighting-gallery.net/gallery/displayimage.php?pos=-228053
You can see the two exhausts of two of the generators of Carmel hospital emits thick black smoke when starting.
Other generators there don't do this, and I think that petrol engine don't do this during starting.
Why this is happening?

Diesel engines usually do this under load?, either theses a slight load on them when starting, or its an automatic cold start feature where more fuel is injected before the combustion starts?

At the beginning of the load test, the maintenance initially starts the engines of the generators and after the engines reaching working speed, immediately disconnecting power to the hospital and than connecting it to the generators.
I hope that @Medved and @Ash know what this phenomenon happening.

Also: in this video ,you can hear at 0:42 a noise of a pressured air releasing that one of the generators emits during starting.
Why this is happening?

Some physical/chemical facts for the beginning.

(1) Motor fuels are hydrocarbons, mainly composed, as said, from carbon and hydrogen.
(2) Diesel fuel has notably higher carbon to hydrogen ratio than petrol.
(3) Carbon rich fuels need more mass ratio of air to burn completely, to clear carbon dioxide and water, as carbon takes two oxygen atoms for on carbon atom, and hydrogen takes one oxygen atom per two hydrogen ones.
(4) Diesel fuel has notably higher boiling temperature range than petrol.

That said. As carbon rich fuels have more carbon and need more oxygen to burn, they tend to start to soot easily when burned incompletely without all the oxygen needed. Then, when engine is first started, it sure runs at sub-optimal cycles. For low RPM, compression will be poor and air dose will be lower than usual. Also, engine does not start immediately and fuel doses injected on previous cycles will partially stay in the cylinders making excess fuel. As diesel has high boiling point, extra fuel will stay for long on the cylinder walls. All that makes for low oxygen and too many fuel, making the engine to soot for some time. Also, when engine is stopped, some oil enters the cylinders, acting then as a fuel and taking some oxygen to itself. Even heavily worn petrol engines can soot because of this.

As for why one engine smokes at start and second don't, these may be from different generations. Older engine may be worn with worse compression and more oil leaking into cylinders. Also older engine might have a mechanical fuel pump which delivers just a fixed fuel dose per rotation, having sub-optimal conditions a start, and newer engine may have electronic common rail. Common rail systems get quite sophisticated, running at about 2000bar (!) fuel pressure for precise fuel atomization and having a powerful microcontroller opening injectors in a programmable way. So, on start, it may wait for crankshaft RPM to stabilize with no fuel, and then begin gradually opening the injectors until engine pick-ups, managing a smokeless smooth start.

It depends on multiple factors:
Too much fuel for not enough air:
How the governor works, mainly how much fuel it injects during start (simple mechanical ones just set full injection dose so it starts and spools up rich until it reaches set rpm, more complex hydraulic ones tend to slowlu increase fuel dose, so unless the cranking takes too long it starts with small dose and spools up before the dose increases, most modern electronic injectrion control is able to inject just the correct amount to not yet create any excessive soot).
Also how the max dose limitter is adjusted plays a role, or even whether it is made to follow the charge pressure (mainly with turbo charged engines, which get way more air pressure when running vs when starting with turbo still stopped, while the fuel dose limit uses to be set to the running pressure, so with turbo spooled up),

Improper burn conditions
Does the engine feature glowplugs and are they working/preheated during the start? Without them the fuel burns too long and is not atomized that well when cold, with the higher injection dose at spool up it leads to incomplete burn.
Slow moving injection pump cam may lead to too low injection pressure, so also improper fuel atomization.

Except the most modern engines, the smoke on start used to be tolerated as it was supposd to happen only for a brief moment.
But once the emission standard tightened, mainly requiring the use of things like DPF, the engine management had to be done so to prevent the smoke even at start (as that would clog the filter way too fast)

As for air release sound, large diesels are often started using compressed air. That saves the need for an enormous starter motor and battery. Air is just stored at rather high pressure and entered in the cylinders in a sequence using special valves. Also there may be some starting aids operating pneumatically.

If It have an mechanical pump, It May be due shutdown method: because diesel doesn't have Spark plug, the only method Is to close aspiration with a valve.
When valve Is closed, It Will run briefly in Rich mixture, so It Will create a lot of dust which Will be deposited on muffler.
At the next start, the dust Will be blown out and then, here the smoke.

@marcopete87: I don't think that dust is the cause of the huge and thick black smoke during starting. The large generator of Shmerling near the trash compactor produces much less black smoke during starting, and the exhaust of the generator for the reverse osmosis system don't emit black smoke at all during starting.

Normally the shut down should cut off the fuel injection first, regardless what fuel injection control it is using (pure mechanical, hydraulic governor, electronic,...; all have an input lever/signal to cut off the fuel for engine shut down). Air intake choke is only a backup method (it is able to stop even a runaway engine in case it starts running on its lubricating oil or so), it is to be activated only with some delay or when the fuel cutoff is not effective.
Plus you don't want the unburnt lightweight diesel fuel to dissolve and wash out the lubricating oil film on the cylinder walls after the shut down, so it is important to stop the fuel first before the lubrication is stopped (so the engine itself spooling down).

So unless the engine is faulty (e.g. misaligned shut off sequence control cutting out the air before the fuel, or the engine is actually burning oil or so,...), there is no reason for any smoke at shut down.

I know that car petrol engines starts to exhaust gases and smoke only after the starting and not during the starting judging by the noise of the exhaust. I've heard this with my father Subaru XV crossover.
Why is this?

In the case of the video it's just an old type of governor. Because at startup the RPM is not at the set value, the engine gets maximum injection and because it's not yet at speed, the air/fuel mixture is incredibly rich causing the smoke.
This can be prevented with basic electronically controlled injection systems, which don't just put fuel to 100% on startup.

A similar situation happens when you have a turbodiesel that you suddenly load up. First the governor opens, and only after the turbo spools up, you'll have a good air/fuel ratio again, and the smoke fully disappears.

Petrol engines should never smoke. If they do, they're burning oil. During starting you definitely get exhaust gases coming out of the tailpipe but no smoke. Water vapor can very much be exhausted, if it's quite humid and cold. but that often goes away within 10 minutes or so.