Don't stress too much about it. You had a bad day, you snapped, you feel terrible about it. Put it down to experience and try to find ways of coping with his stubborness.
We sometimes have similar problems here, as my DH is of the opinion that sometimes it is necessary for the DC to just do as they are told. He can be quite strict with them and fight battles that I find not worth fighting.
I have had the struggles with the seatbelt and had to use force to get it on, but now just sit in the car and refuse to drive unless DS is strapped in.
In a similar way, I have taken DS to kindergarten in his slippers as he has refused to put his shoes on. (when going in the car).
I tend to be more non-confrontational than DH who expects obedience. It is to do with his father, who is was very strict. My parents were similar to my parenting ideas. DH is improving, now that he sees that it is less stressful to talk to the children.
Whereabouts are you in the Alps, Enfield? We are near Geneva. The weather is miserable today.