Could a flight ever be considered to be operating under Visual Flight Rules and Instrument Flight Rules at the same time?
No, you can't be both VFR and IFR at the same time. It would be like saying that a ball is both red and blue at the same time. You are either one or the other, never both.

If the pilot hasn't received an IFR clearance from flight services, then they are operating under visual flight rules, regardless of their intentions, capabilities, or flight plan. The sole exception is in class G airspace, specifically when they're flying a pre-set departure or arrival procedure from an airport. If a VFR pilot enters a cloud (and there's no emergency in progress), then they are breaking the law, and they can't just decide later that "oh, I was actually IFR at that time".

ATC is never going to inquire about which rule set you're operating under unless there's some kind of administrative foul-up. They're the ones that either did or did not issue you the clearance, so they already know which rule set you're operating under.