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This is going to sound quite pithy, but I think it comes down to feeling that your arrangement to homeschool is your due.

I would bring a stack of business cards and, if anyone interrupts your class, introduce yourself briefly as your DD's outside (class X) instructor and say, "We're in class right now. If you'd like to get in touch with me, please contact me at these coordinates." You are offering educational instruction and should expect the same courtesy as any conventional teacher, including not being interrupted in your "classroom". I wouldn't engage in the "are-you-or-aren't-you-a-teacher" politicking. The school has already decided you are your DD's teacher, so third party heresay is irrelevant.

Do you have a designated space arranged for you by the school? If so, a polite, "Excuse us please, folks, could you kindly take your lunch to another location? Principal X (or whoever) has allocated this space to us for private instruction." should suffice. If you keep your messaging polite and consistent, everyone will ultimately internalize that space as another classroom. If you don't have a designated space, I'd probably want to get the principal's okay for a more predictable arrangement, even if you have to move a bit throughout the morning.

Just a question, but could you help me better understand the legal requirements for you to be on school property?

Can you put a sign on the door that says "Class in session. Please do not disturb"?

Well, unfortunately, since you live in a state where partial homeschool isn't permitted, I think that you are going to have to opt for the laying low approach...

Maybe you and the teacher/admins who came up with the plan can meet periodically to discuss things which are working/not with the arrangement-- and mention that you aren't sure how they would prefer you explain this situation to curious onlookers (obviously official and with the school).

Should you refer to this as "I'm helping [Teacher] in the classroom?"

Or something else?

Is there a location where you "won't be disrupting others?"

(Yes, it's galling, that last one... but it is a way of subtly bringing up the fact that your DD deserves to have a learning environment that isn't shoved off in some corner of the faculty lounge!!!)

Maybe your party line can just be to punt it back to the principal? "After discussion with Principal X, this is the best curricular solution for DD given available resources."

The school should be able to ensure that you have access to a table in the library at all times. That's no place for teachers to be eating lunch or gossiping.

What!?! I thought teachers ate in the "Teacher's Lounge". I've never heard of teachers eating in the library and I actually thought food and drink was frowned upon in the library no matter who it is that has it.

I agree there should be a better place for you to go where you are not on display nor interrupted.

That being said - it sounds like at least the principal is trying to accommodate you, but maybe she/he should have informed the staff better. You don't want to burn this bridge, but maybe ask if there is a more conducive space for you and your DD to go.

Is there any risk these teachers are going to report to the district? I very much agree with HKs suggestions re checking in with the principle (and whoever else) regularly and putting this issues in terms of "How would you like us to discuss his with staff?" "Is there somewhere we might cause less disruption?"

Now that its not a secret the principle possibly needs to address gossip head on in a staff meeting, but they'll want to think that's their idea (and of course they need to know its an issue).

How about "no, I am a physics tutor". It sounds like a good arrangement despite the problems.