Was I in the wrong for this minor fender bender?

‌Yesterday, while waiting at a red light on my way home from work, a car behind me rolled into mine with a slight bump.
Since my car hasn't been running well, I initially thought the thud was just my car stalling.
My car is an older model, so the bumper and everything else looked fine, and the other car didn't seem damaged either.
I exchanged contact information, took some photos, said I'd call if there were any issues, and hurried off because I was busy.
Today, my shoulder feels stiff. I already have some shoulder issues, but it feels even more painful, so I'm wondering if it's because of yesterday's accident.
I contacted the other driver to request a personal injury claim (medical coverage) so I could stop by an orthopedic clinic after work.
To be honest, I wanted to settle it privately with 200,000 to 300,000 KRW in cash and be done with it,
but the other driver didn't mention anything about that yesterday, and I felt awkward bringing it up first. Since the car seemed fine, I just said I'd go to the hospital.
Then, they immediately asked if I was going to an Oriental medicine clinic.
I told them I was going to an orthopedic clinic to get checked out, and if necessary, I plan to receive light outpatient treatment for 1–2 weeks—I have no intention of being hospitalized.
I actually have no intention of playing the "patient" card. If I were going to be a nuisance, I would have started complaining about the bumper or the rear sensors being broken.
So, they processed the personal injury claim but then told me they would apply for the Madimo program (accident analysis program).
I told them to go ahead and said let's not waste any more emotional energy on this.
It felt insulting because they treated me like a fake patient. Regardless, I plan to just go to the hospital and proceed with treatment as originally intended.
In the case of a minor traffic accident, is it standard practice to just move on if it doesn't look like a big deal?
And is it wrong of me to have wanted a cash settlement in the first place?