I too have struggled with the concept, but mainly because it uses the word 'equity'.
When I now consider the concept, I think that it is the likelihood I can bet off a villain versus what it might cost me if I'm wrong. I determine the likelihood via reads, PT stats (both his, and what I have been showing him in this game or for the last 20-30 hands), position, stacks, and what hand I hold.
I am primarily a tourney player, and IMO the concept in tourneys is maybe not as important as it is in ring, at least in the early/middle part of the tourneys (largely because of the constant table shifting, with new personnel at the tables). Toward the end, already ITM, ring style play becomes much more prevalent in most tourneys. But a big part of that is that often, in late stage tourneys, I, and possibly my villains, have stats accumulated thru the present game that give more info about the players. So for example, even in late stage play, I might not consider a Fold Equity situation the same if I just moved to a new table with few stats on the players.
Perhaps my discomfort with the word fold 'Equity' is that it sort of comes down to how likely am I to pull off a bluff, semi-bluff, or value bet and get my villain(s) to fold. So maybe it is just a meter or gauge (just like on your dashboard) that tells
me something useful. Maybe 'Value Meter' would be a decent name for the concept. Except that Fold Equity seems to be dealing specifically with the likelihood that you can get your villain to bow out of a hand.