I never actually "kept track." As long as (s)he's moving, I never knew that it mattered how much, or what times of day. It's safe to assume (s)he'll move more when you are lying down or still, though. Think of the movement of your hips when you are walking, as a sort of hammock or swing, for where baby is resting. It's very relaxing, and these are the times baby will be sleeping.
Twenty weeks is halfway there. Technically, pregnancy lasts for 10 months (as measured from the start of your last period: 40 weeks divided by four weeks/month = 10 months). By that measurement, yes, you are at 5 months. If you are measuring weeks from approximate conception, you are technically at 4 1/2 months.
Congratulations!