SQL Server: update trigger fires before insert trigger
Considering that a) Triggers are naturally part of the Transaction that is the DML statement that fired the Trigger, and b) an UPDATE cannot happen on a row until the row exists, it is impossible for the actual UPDATE to show up before the actual INSERT. Hence, something else is going on.

Things to consider:


Are the Trigger definitions shown in the question the actual and current definitions? Is it possible that the Comment values of "Insert" and "Update" are switched in the Triggers such that the INSERT Trigger has the comment of "Update" and vice-versa?
Is it possible that the UPDATE Trigger is somehow defined as being AFTER INSERT, UPDATE ? If so, an INSERT operation would fire both triggers making it look like both an INSERT and an UPDATE happened when in fact there was no UPDATE operation (which might also explain why the times are only a few milliseconds apart for the audit entries).
Is it possible that your query to determine "corresponding" records is flawed and that the results are misleading?