Exactly this!

I'd also point out that thoughts of the past always occur in the present as well.

And remembering something is not like replaying a movie, but every time you revisit a memory you change it a bit based on your current state.

Remembering is an active process.

Yes indeed, including in some cases changing the temporal order of events based on reason or new data. For example, I once misheard a drinking mug shatter on the floor before it slid off the kitchen surface.

But reason corrected the memory.

https://hilo.hawaii.edu/~ronald/310/310-Dennett-MultiDrafts....

I'd like to add that I think 'living in the moment' is powerful and useful despite not being literally true because it points to something real and important.

Well the term 'living in the moment' is just a western paraphrasing of the much more subtle and thorough explanation of mindfulness in the original suttas.