This is where I think there's a fine line, and I know people will be split on what I'm about to repeat (as I have said it in other threads).
If you KNOW your partner has anxieties, jealousies, concerns, etc, then you're between a rock and a hard place.
If you tell your partner everything... "I'm chatting to 'Clare' right now... nothing special, just Wolverine vs. Deadpool" - i.e. it's a shared interest (innocent), but not something your partner is into... then 10 minutes, an hour, a few hours while you're casually watching a movie with your partner on the couch, is no biggie. You're not going to fancy this person, it is JUST a mutual interest.
The problem is that if you tell your partner, and they have these 'issues', it WILL turn into an argument. You know you have no intent, you know they have no intent, and it is just a friendship... but your honesty WILL result in jealousy and arguments. OP looked through his phone "Things they were both interested in going to see." <--- so similar movie interests.
Now, if he doesn't tell her "he lied about her being there too that evening as it would have upset me", he flips a coin. Telling her WILL result in arguments, but not telling OP (about movie discussions, her being there (justifiably) at work events), will be peaceful. Of course, if OP finds out, "the fact he lied hits hard" - then MN pile on and say hiding it means he's guilty, get your ducks in a row, etc...
He's in a lose-lose. Of course, I appreciate that there's a fine line between hiding because you are doing something, and hiding so as not to face an argument for even talking to a friend... but I think most people would opt for taking the chance.
People do it all the time... the 'lie' to avoid arguments... "you're the best I've ever had", "size doesn't matter", "I haven't been with many", "we're just talking about shoes (when you're actually complaining about your DP)". How many women here have guys that they have friend-zoned still as friends that they still chat to?
I honestly think this is dangerous territory. Don't get me wrong, it could be an emotional affair, but I don't think it is (yet, at least). But if OP keeps giving ultimatums, keeps making a big deal out of it, listens to some of the above about 'making him jealous so he understands', it's going to become a self-fulfilling prophecy.