Yes, I agree, that is the main issue and if you can get clear confirmation that that is how the company would handle it, I think you would be in the clear.

If you cannot get that clear confirmation, out of safety I might just enroll assuming that your choices will apply for rest of this year and into next year, and that you will begin your new job's health plan as soon as you start work.

Now here is a bit of gray area up to your discretion/judgement: You might try enrolling in COBRA anyway and submitting claims under your old plan. As you said, you're not supposed to have COBRA once you have another health plan, but maybe the systems are not in sync to tell each other that very well, especially for a new employee enrolling in the middle of a month towards the end of a calendar year. Maybe the worst that could happen is that an expense/claim gets bounced back and you have to claim it under your new plan. (and you would be out the COBRA premium as it didn't end up being used as you thought) It all depends on whether the cost/risk of doing this is worth the difference in out of pocket expenditure you're saving.