Yes as I showed in the screenshot in the post right above there's the begin xact and commit xact log records (which would be your two per cycle).

This is what's is in the log after that test of 10 updates:

The update itself is not logged. If it was, there would be a LOP_MODIFY_ROW for each update (10 of them), which there is not.

(btw, you'd get the begin and commit even if there was no explicit begin and commit, as all modifications are in implicit transactions)

If I change the query and update name to "Bob", the log records look like this

That Lop_Modify_Row is the log record for the actual update, and it is missing in the case where the row is updated to itself, hence we can conclude that the update is not logged when updating the value to itself.

The question was not, does the update cause any log records to be written to the log. The question was, does the update dirty the page if the update makes no changes. It does not, as I showed with the DMV previously.