What are the different types of procedures for starting a fuel injected engine?
The key thing that makes FI engines different is how you prime them.  You are using the pressurized fuel injection system itself to do the priming instead of pumping fuel into the intake ports using a syringe pump primer.  You do this by turning the boost pump on to pressurize the system and setting the mixture to rich, which on a crude-ish mechanical FI system like the continuous flow systems used on airplanes, just starts spraying fuel in the injectors as long as the mixture is in, whether the engine is turning or not.

Two main differences from a carb:


The priming procedure is one of timing using mixture, not counting pump strokes.
You start the engine with the mixture in idle cutoff (because with the system pressurized and the mixture not at idle cutoff, it'll be spraying fuel in continuously and you'll soon have a little pond of fuel under the cowl).  You don't advance the mixture until the engine catches.


As far as hot/cold starting goes, the basics are the same as a carbureted engine, more prime when cold, less prime when warm, no prime when hot.  So for a cold engine that uses 6 strokes of a manual primer, with the FI engine it's pump on and mixture rich for some period like 5 seconds, then back to idle cutoff before you crank it.  On a warm engine that uses 2 strokes, the FI version might be 1-2 seconds.  You're doing the same thing, just with a different process.

Because it requires careful timing with prime and then keeping the mixture at idle cutoff until start, it's easier to screw it up and flood the engine by messing up the timing, or to be late with the mixture after it starts and it goes and dies on you.  So they can be a little bit fussy until you get the hang of it.   If you're bush flying on floats in something like an injected C-185, where you need to be able to know you can start right now once you push off from the dock, getting it right really becomes important and hilarity will often ensue (drifting toward rocks or moored boats in the breeze as you scream and curse) if you don't.

At the other extreme, because the injector distribution system is normally on top of the engine where all the heat collects, when an FI engine is hot on a hot day it can cause vapour lock in the fuel distributor and lines (even with the very low Reid Vapour Pressure of avgas) and they just don't want to start at all until the cowling cools down a bit.  Not realizing this, you may decide that it needs priming, so you start priming when it doesn't need it and things have cooled down enough for fuel to start to flow again, and now it still won't start (the raw fuel smell with be a clue).  They can be frustrating beasts.