I never liked that notation for that reason, it can confuse some people because it is sort of a misnomer. To me start up should only be what leads up to the first active frame. That other notation has overlap where what they are calling start up includes one of the active frames.
I think of it simply as, if a move has a 3 frame startup, it can’t punish things that are -3 frames or less, even if it’s a frame perfect punish, because the move isn’t active until the 4th frame.
If I refer to something to a 3 frame jab, is means it has 2 frames of start up, and is active on the third frame, thus it can punish things are -3 or worse.
This really isn’t conveying any different information than the posts above, merely rephrasing it a bit.