Raising a spectre puts it into your spectre pool, a list of mobs that can appear when you use desecrate. Slain spectres won't auto-revive, you'll need to spam desecrate until the corpse shows up, then raise it again (you can do this in hideout or wherever).

Someone else can "give" you a spectre by using their desecrate, then you find the corpse in the pile and raise it.

The corpses desecrate spawns will be chosen from the current area + a few from your spectre pool. You might need to use desecrate multiple times to find the replacement for the exact spectre that died.

Yes, this is is annoying - many spectre builds go to great pains to ensure their spectres don't die.

If I ask someone to Desecrate Carnage Chieftain and I use Raise Spectre to capture it

go to southern forest in act 7; you can find both apes there iirc. once you've summoned it, it will be in your desecration pool, and you can cast desecration to get another one to raise

Or can I simply press summon spectre again and it’ll resummon it by itself ?

don't do this because you'll spectre something random and remove one of your spectres from the pool

does Desecrate now pop up 3 types of mob randomly ?

desecrate summons a random selection of zone mobs and whatever corpses you have spectre'd.