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 the 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.