As long as the edit looks legitimate (it still makes sense with the question), it should be okay to accept the edit anyway. That way, if it turns out that the edit was not done by the same user, it can easily be rolled back by a moderator. But if it's still rejected, then the OP will have to make the edit with the original account to ensure that it's done.
It turns out that the users are the same, so I've requested an account merge. As such, do not be alarmed that the questions shows a suggested edit from the OP; I assume that's by-design.