How to tell what invoked a trigger in SQL Server?
Couple of Options:

Option1:

Your best option is to run a server side trace with filtering options enabled.

This way you can narrow down the amount of data captured.

Option2:

Depending on the version (Enterprise), you can use SQL Audit as it allows you to track DELETE, EXECUTE and other Actions on the Object level as well. So you can filter the table that you are interested and enable the audit. Note that SQL Audit behind the covers
uses Extended Events. 

Option 3: (I have not used this .. but its from msdn)

Light weight SQL Server procedure auditing without using SQL Server auditing version 2



Another option would be to use CONTEXT_INFO as described in this answer. I dont see if this will help you as you will not be modifying legacy code.