Sort of. Oral HSV-1 can spread to genitals during oral sex. The article notes this is becoming more common.

As for HSV-2:

  -More than 80% of people with herpes are undiagnosed
  -Overall prevalence by the time people reach their forties is 26%.
  -In the general population, one-fifth of women and 11.5% of men are infected.

http://commonhealth.wbur.org/2011/04/latest-genital-herpes

You are right. HSV-2 is the infection mostly occurring in the genital area. However, HSV-1 (cold sores) can also transmit to the genital area for example via oral sex, even though such HSV-1 infection there is less likely than with HSV-2.

Addition: while oral sex often goes unprotected (HSV-1), HSV-2 transmits also with condoms during sexual penetration. Skin-on-skin contact in unprotected areas is enough for the virus to transmit. However, because of the motion and friction condoms protect women better than men.[1]

[1] http://www.guttmacher.org/pubs/journals/2721301b.html

The CDC considers Herpes Simplex to be an STD if it involves the genitals [1]. According to the article, both HSV-1 and HSV-2 can cause genital herpes.

[1] http://www.cdc.gov/std/herpes/default.htm

There are two common strains of the herpes simplex virus, called HSV-1 and HSV-2. Both can cause genital sores, and both can be transmitted sexually. Here are some facts: http://www.who.int/mediacentre/factsheets/fs400/en/