The original syllogism

All men are mortal.
Socrates is a man.
[Therefore] Socrates is mortal.

This fits the form:

All A are B
C is A
Therefore, C is B

Yours

All syllogisms are arguments from major to minor premise
Socrates is not a premise
Therefore, "All men are mortal..." is not a syllogism

seems to be of the form

All A are B
C is not D
Therefore, E is not A

Does this make sense?