Can a 403(b) that is not subject to ERISA in operation - meaning it meets all of the requirements - limited employer involvment, no employer contributions, etc. but which has a document that subjects it to ERISA claim NOT to be an ERISA plan?

It won't be subject to ERISA, but it will have to apply ERISA to itself if it incorporates ERISA by its terms. Then you have to look at the terms of ERISA that got actually incorporated. You may find a few paradoxes, such as the ERISA requirement to file Form 5500, but ERISA only requires filing of Form 5500 if the plan is subject to ERISA. One wonders if ERISA permits filing of Form 5500 if the plan is not sbject to ERISA. This should you you busy for a while.

Or you can contemplate whether or not the adoption of that plan document by the employer is sufficient employer involvement to cause the plan to be subject to ERISA. I think that is a viable position because adopting an arrangement that that goes beyond being a mere conduit (by imposing ERISA terms) is much more involvement than necessary.

This, of course, assumes that the employer is not one of those exempt from ERISA - that is, not a church/QCCO that did not affirmatively elect ERISA coverage, or not a governmental employer (such as public education).