T-sql – How is a T-SQL Trigger that only fires on real changes made

I have a table trigger on UPDATE and INSERT that adds a row to another table. It only needs to add a row if one of four columns are changed. I tried using IF UPDATE(col) to test for changes but it has a blind spot. It only tests that some value came in. I need to go deeper, I need to compare the old and new values to see of a true change has occurred. It has to work with both INSERT and UPDATE.

In the case of an UPDATE that's easy because both the inserted and deleted tables have values I can compare within the trigger. However, for the INSERT only the insert table has values. Because I need this all in the same trigger, how do I handle that INSERT case?

Here is the script of the trigger I want to modify:

ALTER TRIGGER [dbo].[trATPerson_alter] 
   ON  [mydb].[dbo].[AT_Person]
   AFTER INSERT,UPDATE
AS 
BEGIN
    SET NOCOUNT ON;

    -- Not all updates require a push
    IF (UPDATE([First_Name]) OR UPDATE([Last_Name]) OR UPDATE([JobCode]) OR UPDATE([Inactive]))
    BEGIN
        INSERT INTO [mydb].[dbo].[AT_Person_To_Push] (
                [Facility],
                [VendorID],
                [Person_code],
                [First_Name],
                [Last_Name],
                [JobCode],
                [Alink],
                [Inactive]
            )
        SELECT  [Facility],
                [VendorID],
                [Person_code],
                [First_Name],
                [Last_Name],
                [JobCode],
                [Alink],
                [Inactive]
        FROM inserted 
    END
END