Let me simplify, Wings
To start a turbine, you basically spin it up (using a starter) to about 25% of max rpm. Fuel is introduced, ignited, and the engine spins up to idle. A fuel control unit regulates engine speed. This is essentially the same for a turboprop or jet.
Jet starters are usually air-powered, and this air (compressed) usually comes from an on-board APU (a small turbine in itself). The jet is relatively easy to start when everything's working right.
On the Twin Otter there is no APU (I think), and the starter is electric. With no engines running, the battery powers the starter. If the battery is old and weak, you'll get a "poor" start; e. g., the turbine may not accelerate properly and start temperatures may rise to too high a value. In any case repeated battery starts are hard on the battery. With one engine (and its associated generator) running, the load on the battery is much less and there's more current to turn the starter.
A "hot-start" is like the "low battery" scenario above; usually you can see the slow acceleration and rapid temperature rise and cut off fuel before the situation gets serious. It has nothing directly to do with a quick turnaround. "Hot-starting" a turbine - a term you used above - really isn't proper verbiage unless you're referring to said malfunction. It's certainly proper when referring to, say, a fuel-injected piston engine (as in starting shortly after shutting it down).
PP's explanations are good; mine might be a little easier to follow. Read theirs after mine and things will make pretty good sense (I hope).
