I believe that this scene actually can have one of several characters, depending on an arcane romance tracking point system?
Yep, as others have said, there's four possible dates, with all three girls as well as Barret. I wanted to put a link here so you could check out the byzantine system that decides who the date is, but as it happens, the wiki in that page has a spoiler for
the very end of the game, which I'd totally forgotten about. So, please don't go check that out by yourself.
Overall, Tifa isn't that hard to get - you just need to blow off Jessie at the beginning, always choose the options that prioritize Tifa or her interest when she comes up (at the bar at the beginning, for example, accept the drink when she offers it, say that you'd never leave her instead of apologizing, flirt about sleeping next to her; claim her as your girlfriend when Aerith asks; when you fall into the sewers, check on her first; when situations like Marlene asking if you like Aerith or being given the chance to express jealousy of Aerith's previous boyfriend, never do so; pick Tifa but not Aerith whenever the game specifically asks "which team are you forming?", such as when saving Red XIII or leaving Midgar; disagree with the woman in Kalm that says Mako energy is good actually; etc), and you'll get her date legitimately. It's not necessary to use the bug of telling her "I'll get us out" in the prison 255 times to get Tifa as a date, it just takes being
consistent about always preferring her.
Yuffie, on the other hand, you can only get by specifically gunning for it with pre-knowlege, since you need to repeat her recruitment four or five times near-perfectly and then specifically pick all the options that reduce the other girls interest, most of which are out of character for Cloud; if you recruit her first try, then her date isn't happening. Barret is possible to get by accident because, while he starts with the lower score, there are a number of situations where failing to do the thing that earns points for either Tifa or Aerith defaults to handing points over to Barret, and he also has a few situations where he earns points just by talking to him and saying things he agrees with, which mostly fit Cloud's character and thus are likely picks. It's still hard, because it requires lowering Aerith's score as I mentioned before while also lowering Tifa as much as possible, but it can be done - especially because doing the correct rendition of the Wall Marked section (the one where Cloud is picked by Don Corneo) and then answering Corneo's question by mentioning Barret actually gives him a solid boost.
You honestly did bad enough with the barrel minigame that I suspected you'd manage to get Tifa, but after you brought Aerith along for Barret's backstory (that's actually one of the most important events to determine closeness with Cloud) and didn't get Tifa's unique dialogue in Gongaga, which she only has if you've earned enough points with her, I guessed it'd end up being Aerith in the end.
I would suggest looking at all four dates though, if only for the fun of it - you can probably reach them all from the link that appear at the end of the Barret date video that was posted:
The Yuffie date especially is worth watching since it's, in my opinion, even more out of left field than Barret's in how it plays out.