Wrong commit user

I have 2 accounts registered with 2 different usernames but samen e-mail account for administration reasons. I have 2 different generated SSH keys and assigned them to the 2 accounts.

I made an inital commit with user A on system A (git config is set up for user A)
And worked on a different system B (different git config for user B). When I commit on system B and push it to Bitbucket it will show up with the wrong username (which is A) in Bitubucket. When I do a git log on system B it says it is commited under user B, but on Bitbucket it is commited under user A.

This is probably due to the same e-mail adress? I guess the committer is uniquely identified by the e-mail adress? Which is wrong I guess... It should be recognized by the SSH key or just simply used from the "user.name" setting?