It's a wonderful thing that you took it upon yourself to learn more about hpv! Doctors give their patients so little information on the subject. What's worse is that doctors HAVE so little information on the subject. I will try to answer your questions and I hope that having more information will give you a sense of relief concerning the matter.
First of all, it's totally normal to have acquired an hpv infection. Dr. Hunter here on MedHelp stated in one of his posts regarding hpv that we ALL acquire genital hpv infection. Most people do (except in the case of two virgins coming together and staying together for life). The reason so few people are aware of their genital hpv infection is that hpv doesn't usually cause symptoms.
There are many many strains of hpv. Some are restricted to the hands and feet while others are strictly genital. You said "I know there are four types. Two cause genital warts, two cause cancer." I believe you are referring to the 4 strains that the vaccine may protect against. There are actually 30-40(ish) strains of sexually transmitted hpv. Low risk strains of hpv may cause genital warts but usually don't. High risk strains of hpv may cause cervical cancer but usually don't.
Routine pap smears and even std checks do not test for hpv. An hpv test will only return a positive result for an ACTIVE infection. Women under 30 are usually never tested for hpv because it is expected that they will be infected and that the virus will soon be cleared from their system without complications. Even if a woman under 30 has abnormal cells, they are still unlikely to actually be tested for hpv because it is assumed that hpv is the cause of their abnormal cells. Men are not tested for hpv and the only way a man will ever know he has an hpv infection is if he has visible warts. In your case, you can be sure that you and you partner do in fact share this strain of hpv that has caused abnormal cells for you.
There is a lot of controversy regarding whether or not hpv infection ever really does go away at all or if it simply goes dormant in the body. It probably does stay in your body forever, like a cold virus, which would be why you can not be reinfected with the same strain of hpv. However, hpv can reactivate later on, all the more reason to make a lifelong commitment to healthy living.
The immune system is responsible for keeping an hpv infection in check. That being said, try to lead a healthy lifestyle! Do not drink or smoke. Eat healthy, lots of fruits and vegetables, and restrict your sugar intake. Make sure you are getting an adequate amount of sleep at night (8 hours). Exercise and take your vitamins. Also, try to relax as stress wreaks havoc on the immune system! If you are interested I have a book recommendation for you- 90 Day Immune System Makeover by Janet Maccaro. You can buy it very cheap on eBay and Amazon.
I hope this has cleared up some confusion for you. Let me know if you have any more questions regarding hpv :)
And even if I don't have genital warts, could i give it to my partner?