Reaching round two means they believe you can do the job. The question has changed.
Bring new material
Repeating your round-one stories to a second interviewer wastes the opportunity, especially since interviewers compare notes. Prepare a different set.
Have a view
By this stage you know enough to say something about their situation. "From what I understood last week, the bottleneck sounds like X — is that right?" separates you from candidates who are only answering questions.
Expect the practical questions
Notice period, salary expectations, start date. Have real answers ready rather than improvising numbers.
Common questions
How many interview rounds are normal for remote jobs?
Three to four. Remote employers often add a round to compensate for not meeting you.
Should I repeat my best story?
Not to a new interviewer. Interviewers compare notes and repetition looks thin.