Move records to a different table using MERGE in SQL Server ?

MERGE can do any combination of these things "simultaneously":

INSERT INTO tableX
UPDATE tableX
DELETE FROM tableX

It cannot modify two tables at once. It cannot do these "simultaneously":

INSERT INTO tableX
DELETE FROM tableY

So, you have to use two statements, one INSERT and one DELETE. Your original queries are fine. Using an output ... into clause to your delete could enable you to have better perfomance by using a join on a primary key instead of a where. Still, that table representing students, I doubt the table size is big enough to justify (even this little) extra work.

Of course, you can also use two MERGEs instead, but this is not just dangerous, it is stupid.