Unsure how to handle staying in touch with someone I still care about

So I (26M) was seeing this girl (25F) for about 2 months. Things were good, but she felt we weren’t on the same page emotionally. She said I wasn’t as ambitious as her (though I am, maybe just in a different way) and that we just see things differently. On the last day we met, I communicated poorly and I think that also played a role in things ending. She mentioned she doesn’t see a long-term future with me right now, but maybe if things align in the future, who knows.

When we last talked, she said she didn’t want to be in a relationship for now, but also that she doesn’t want to lose contact completely. I told her my “door will always be open.” Since then (been a week), we’ve been having some casual, light conversations. Sometimes I start them, and she’s been reciprocating well.

The thing is, I still really like her, and I’d really like to get back with her if that were possible. Part of me wants to stay in touch casually and show I care through actions, but another part of me worries I might be hanging onto “fantasy hope” instead of moving on. I don’t want to chase too hard, but I don’t want to disappear either, since she did say she wanted to keep casual contact.

I guess my question is:

How do I balance staying in touch without coming across as pushy or desperate?

Should I keep talking casually and let things flow, or step back more to protect myself?

Has anyone been in a similar situation where it actually worked out in the long run?

Any advice would help. Thanks.

TL;DR: Dated a girl for 2 months, she ended things saying we’re not on the same page but wants to keep in touch. My bad communication on the last day we met also played a part. I still really like her and would like to get back together, but I’m unsure if staying in casual contact is healthy or if I should step back. How do I balance this?