Assumming that you have a single trigger on the table to handle inserts, updates and deletes there is a way to identify what caused the trigger to fire
Use the tables inserted and deleted.
Where there is only a row or rows in the inserted table it was an insert that fired the trigger
Where there is only a row or rows in the deleted table it was a delete that fired the trigger
Where there are rows in both inserted and deleted it was an update that fired the trigger