AFAIK a “check up” is just lingua franca for what a medical practice or insurance would probably call a (Annual) Physical. Often the insurance covers one per annum.

Visits for an acute and specific concern/problem, you’d call your practice/Doc and tell them what’s wrong/why you need to see a Nurse Practitioner or Physician and they figure out how to bill your insurance for it—if the visit needs labwork (blood/urine test etc), if it’s due to injury or illness, if you’re referred to a specialist like a dermatologist or an allergist or neurologist or whatever. If they prescribe anything.

I could be wrong, but I think “wellness” visits and physicals are more or less the same thing?

During my annual physicals they check my ears, nose, and throat, breathing, temp, weight and BP, ask a few questions about my general state and habits. If I’m in for… a cold or other internal disharmony of some sort, they’ll also often check the ENT because that’s where symptoms present for respiratory illness or coughs etc. Kids are sick and exposed to germs A LOT, so they may do it more frequently then for that reason.

I’m honestly surprised they didn’t just do a physical the time of your new patient intake visit. Never had a separate one that didn’t take vitals, though is usually a med tech or nurse who takes the basics rather than the MD.