I might name myself as a tutor or lecturer but not as a teacher.
What does "not" modify?
Does "not" modify "as a teacher"?
I might name myself [ as a tutor or lecturer ] [ but not as a teacher ]. You are right. This is a coordinate construction where the two bracketed elements form a coordination of preposition phrases, the second of which contains the modifier "not".
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I might name myself [as a tutor or lecturer] [but not as a teacher].
You are right.
This is a coordinate construction where the two bracketed elements form a coordination of preposition phrases, the second of which contains the modifier "not".
Incidentally, your example is not a natural way of putting things. You might consider saying: