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I'm sorry, I don't know what choices you made but Baxter is always meant to say quite directly that he's only a suitor for the season when he asks you out and it's essentially a fling. He shouldn't ever ask the MC out as though it's a normal relationship. If you missed the conversation about that perhaps there was a bug where it got skipped.

But I'm glad you enjoyed his story!

A fellow recent player of Baxter's DLC here to address the spoiler question/concern as best as I can:

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So it might play different depending on dialogue choices and how direct/shy your character is set to be towards Baxter (this creator does an AMAZING job making every playthrough unique if you choose different traits and options), but I do know that for my playthrough (direct crush stage with him, with an MC who leans towards playful banter and direct statements) Baxter says upfront that it's only going to be a summer fling. Quoting from that scene in my save file: "Well, before I get too carried away in excitement, I must own the fact that this is not a grand romantic overture. I doubt you would believe me if I was claiming to be your one and only for all time. I am nominating myself as a suitor for the season. A charming summer adventure: no more, no less. I don't care what label you'd choose to put to this arrangement. I could be your boyfriend, or nothing at all." 

And then it sort of does develop-- or can be allowed to develop-- where MC really catches some serious feelings and it's not just a fling for them after all. MC does tend to refer to Baxter as something more than a mere fling, giving friends and family the sense it's a more serious thing than it is. And Baxter can (potentially?) hint he's feeling more than a fling too. But given how he sees himself and how his character is (beautifully) written, just because he might want something more doesn't mean he will let himself have that.

I'm not sure I know what you mean as far as the first kiss and reactions (or lack thereof) and how that translated to their intimacy. It was described as 'chaste' and 'tender', and afterwards Baxter "laughed quietly behind his hand. There was no chance of you making Baxter shy, but there was a definite coyness to his temperament, and a faint flush to his cheeks." (again, just copying from my save file, and there's every possibility it plays differently to different MC attitudes). To me, he came across as just a smooth 'player' (in the most innocent and childlike sense) who rolled with the punches from a forward MC. But I do know what you mean that the exchanging of keys can imply there was a much more intimate relationship happening off-screen, I suppose!