The lesion is called molluscum contagiosum.
Molluscum contagiosum is a common viral infection caused by a poxvirus which can be transmitted sexually or from contact with fomites (objects). Most people with the infection do not have noticeable symptoms although sometimes the lesion is a painless bump which may itch or become irritated. Usually a molluscum lesion will heal without treatment.
Molluscum is transmitted primarily through direct skin contact with an infected individual. Fomites have been suggested as another source of infection e.g. from bath towels, tattoo instruments, and in Turkish baths.
The molluscum contagiosum virus produces a papular eruption of umbilicated lesions which are discrete, smooth, and dome-shaped. They are skin-colored and contain a white, waxy, curd-like core.
It is a self-limited disease, which, left untreated, will eventually resolve on its own.
If the lesion becomes bothersome, then it can be removed by a dermatologist using cryotherapy, a scalpel, or chemicals.
In general, you can contract most STD's from urine, saliva (spit), semen, vaginal secretions, blood, and breast milk.
It is unlikely your sore is herpes because your sore is painless and skin-colored whereas herpes lesions are tiny painful fluid-filled blisters with a red area under the blisters. Although some people don't have any symptoms with herpes, you are asking about a specific lesion you have acquired recently. No, your sore isn't a herpes lesion.
A single firm, round painless chancre could be the primary stage of syphilis. It appears at the spot where syphilis entered the body and reqires direct sexual contact with a person who has an active lesion. The time between infection with syphilis and the start of the first symptom can range from 10 to 90 days. The chancre lasts 3 to 6 weeks and it heals without treatment. However, if adequate treatment is not administered, the infection progresses to the secondary stage. T pallidum (the spirochete that causes syphilis) is a fragile spiral bacterium which can survive only briefly outside of the body therefore syphilis transmission almost always requires direct contact with the infected lesion(s).
Serology and darkfield microscopy or immunofluorescent staining is used to diagnose syphilis. Treatment is a single dose of benzathine penicillin G 2.4 million units intramuscularly.
You have a typical molluscum lesion since your lesion contains a core of a pus-like material.