Do you have to use the word “date” for it to be a date?

Was the below a date, or if not, what would you call it?

I met a woman at a friend’s birthday party, and we had a good conversation. She mentioned she’s really into indoor rock climbing and said I should join her sometime at the gym where she’s a member. We exchanged numbers.

I texted her later and we set up a time the following Saturday to climb together. After climbing, I asked if she wanted to grab coffee and we talked for about 30–40 minutes. (Just for context, I paid for the coffee/pastries, as well as my own climbing ticket. As a member, she didn’t have to buy one.) We hugged goodbye and left.

Neither of us ever used the word “date,” and it felt a little ambiguous.

Would you have considered that a date, or just two people hanging out? If you wanted it to clearly be a date, would you explicitly say so when making plans? How would you have worded it in that case?