If you purchase a home, put down 20k, sign a purchase agreement and register with the city on a land contract for 3 months

Generally, the time period begins when you actually start using the home as your residence.  The home will have to have been your home for at least 2 years during the past 5 years for you to qualify for the capital gain exclusion

So from the moment I lived in the home even on land contract it still counts toward the 2 year requirement. I was curious if it only counted if the house was mortgaged in my name.

The IRS will look at your address, voting records, things like that for your residency period. You have the two years from living there.