The most important are:
HSV1 & 2
HIV
Hepatitis C
These are virus that you are stuck with for life. All the others can be cured with medicine. Of course the others are not good either, but they are not life changing. Viruses are the ones you have to worry about. The above 3 are the ones I always request. If they want to do the other typical STI tests, then that’s a bonus. I hope this helps!
Your partner is pretty low risk for infections. He's had very few encounters, with only 2 partners, and all were protected.
If he wore a condom, he is not at risk for HIV, gonorrhea, chlamydia and trich. The condom protects him from that. There is a very small chance that he would get HPV or herpes while using a condom.
If you'd like him to test, he can go to his doctor and get an HIV test, which is a blood test. He can ask to be tested for gonorrhea, chlamydia and trich, which is most often a urine test, but may be a swab test.
For HPV, there isn't any reliable testing for men. If it's been more than a couple of years, and he hasn't had symptoms, you can assume that if he had it, the infection is gone.
For herpes, he can ask for a type specific IgG blood test, though in many locations, it's hard to get, so don't be surprised if it is where you are. If he hasn't had any symptoms, I wouldn't worry too much about this.
I'm not sure if it's a language issue, but if you meant to use the word "pure", that's concerning. Your partner has had sex a few times. That's okay. If he has an STD, that doesn't make him "dirty". I don't know if you are shaming him for having sex in the past, but please don't. If it's just a translation issue, my apologies. You've asked questions about this before, and your partner is really low risk.