I have no idea what the official name of it is...
Its always the "slow dance" to me

. Where you described you put your hand on her waist while you hold her hand in yours away from the body. This could also lead to where she puts her arms around your neck and either lays her head on your shoulder or you guys make out

.
But from your description you could ballroom dance from the description you've proved I think.
If you're old enough and in the city, see if you can find like a swing, tango, salsa, cha-cha club and bring her there. I believe they give free lessons early in the evening with skilled partners. Then once you get the hang of it you and your date can dance the night away.