I couldn't let that be the end of it.
OP has already gone further to try and not just cut off the 'friend' than I think I would go. It is now up to the 'friend' to make the next move. If that is an invitation to a meal, or even just for a coffee, but fully paid for by 'friend' then you know how the new rules work.
Having said that, the explanation doesn't ring true to me and I would't hold my breath for an invitation. Most of our friends do the take it in turns thing, but others obviously do it differently, so we are used to listening out for clues. Was caught out recently, when people we haven't been in touch with for years, where we used to eat in each others homes, invited us to their new place, mentioning a restaurant where they are regulars, they always take people out instead of eating at home. It sounded as if were were to be their guests, and this seemed confirmed when we arrived. They were greeted as regulars, they asked us what we wanted and related that to the waiter and chose the wine having asked our preferences.
These are the clues to me that they are hosting, but when the bill came, they both mentioned that they had drunk more of the wine than we had, which is obviously a how are we going to split the bill gambit. We were happy to pay our half and immediately produced cash, could just as easily given another card and asked the place to split the bill across the two cards.
That was quite a long story, whose point is to say that even if OP's 'friend' had thought that OP was hosting, she didn't react as if there had been a misunderstanding, she reacted to force OP to pay.
For example, if it were a misunderstanding, and she didnt have any means to pay, 'friend' could have said, oh sorry, I'd forgotten that we share the bill, could you sub me this time and I'll pay for you next or even I'll do a transfer, can you give me your details.
There is so much scope for misunderstandings, it is not surprising that they happen often, so what matters is how the misunderstanding is handled when it becomes clear. In OP's case, the 'friend' showed no intention of paying this time round, so I would wait for the next time, to see how it went, but not hold my breath for that return invitation.
Unless, of course, in a massive drip feed, OP forgot her purse last time and promised to pay the next time....