Is this man flirting or just a polite brit?

Hello! I’m sorry to bring some dating stuff in here but I moved to the UK a year ago from Los Angeles and as a born and raised Mexican, I am noticing I have a lot of cultural differences when it comes to this topic.

I met a man at an art opening a couple of weeks ago. We had a long conversation, that I think we both enjoyed. Afterwards we went for dinner with some friends and he asked for my number.

He texted the next day to invite me to the movies. We went to a Sunday matinee function of In the Mood for Love. Met for coffee before and had lunch after. He kindly paid for everything and chatted a bit over text after I left.

Then there was silence for a couple days. I was traveling so I sort of forgot. But he recommended a book, so I checked-in when I finished reading it to say thank you for the recommendation and send another Sunday matinee. He sounded happy that I read the book and bought us tickets for the movie. For lunch, I offered to pay but he said a split was good.

In neither of the occasions did he try to make a move or anything of the sort. Although, I did notice that whenever I touch his arm to say something he looked at me in the eyes in this sort of deep look.

We are meeting tomorrow to see a ballet that we discussed last time we met. As it was I who brought it up and since he paid twice, I said I’d get the tickets this time (it’s not an expensive function either). He said he was excited to see it and made a reservation for dinner before the function.

As I write I feel like I sound stupid, but I am uncertain if he is just a shy british boy low-key hitting on me or if is he is actually just a gentleman who is happy to make a new friend?

Can any of the British people in this forum help me decipher this behavior? Is this friendly polite or have these been dates? Lol

Thank you xx M

UPDATE: This is no longer a safe channel. So I have to wait a couple days for update. Things are looking good though (IMO)