Is it OK to assume that fast-forward means all commits are replayed on the target branch and the HEAD is set to the last commit on that branch?

When you try to merge one commit with a commit that can be reached by following the first commit’s history, Git simplifies things by moving the pointer forward, because there isn't any divergent work to merge together—this is called a “fast-forward.”

For more: 3.2 Git Branching - Basic Branching and Merging

In another way,

If master has not diverged, instead of creating a new commit, Git will just point master to the latest commit of the feature branch. This is a “fast forward.”

There won't be any "merge commit" in fast-forwarding merge.