From a "light twin" perspective, I've mulled over at least five different starts....for different aircraft....
Left first - because you can see the prop area is clear from the pilots seat...
Right first - because it's nearer the battery (aircraft specific)...
Left first - because it's furthest from the passenger doors and emergency exit (aircraft specific)...
Right first - because it's furthest away from the passenger doors and emergency exit (aircraft specific)
Alernate first start - because it then allows you to listen to the noise of each individual engine, at different times, without the noise of the other one running...
We have yet to mention the idea of "because it says so in the POH".