Mostly I'm thinking about how older cards used to say a spell had to be "successfully cast"....which I thought meant resolve. I just want to make sure I CAN use Momir Vig's first ability whenever I play a green critter.
'Cast' means the process of moving a card or copy of a card onto the stack, choosings modes, division of damage, targets and so forth, and paying it's costs. What older cards used to say doesn't really matter. As far as I can make out, the 'successfully' bit was in case you cast a spell illegally and had to take it back. It certainly had nothing to do with resolving.
EDIT: Apparently I was wrong on that last point. Hey, I started in 2002, I'm a comparative newbie!
Casting a spell means putting it on the stack, making all the required choices for it, and paying its costs if any. Once that process is done, the spell is considered cast, and before players would get priority to respond to the spell, abilities that trigger "Whenever you cast a spell" trigger and go on the stack.
The old pre-6th edition "successfully" wording, if I recall well, really made it so that the triggered ability didn't happen (those didn't use the stack back then, anyway nothing did since there was no stack). There was what was called the "Interrupt window" that would open right after each spell each was cast, allowing other players to respond with Interrupts such as counterspells, and the "when you successfully cast" abilities would happen only if the spell "passed" that window (I think).
I'm a lvl 1 judge, but that doesn't make my answers more valid than those of people who aren't judges. What matters is we can quote the rules to back up our answers. When in doubt, ask for such quotes.