I treat running tights as pants.
When it is warm out, I typically wear running shorts or half tights. Only other time I wear full length/capris tights is when my skin has seen enough sun. Or if I don't want to be wearing short shorts to the store, I will bring a pair of tights to slip on.
During the winter months (seems like 9 months out of the year here in New Yorkistan) I'm wearing full length tights, with no shorts over top.
Never quite understood the shorts on top of pants or running tights... but for kicks, a couple years ago I slipped on a pair of shorts over my tights, and it felt awful!! Never again! It was like I was wearing two sets of underwear with pants and then shorts over top of that. It was just weird feeling...
Although, I think where most people get confused with tights and compression gear is the underwear style tights, versus running tights. Compression tights that UA or the Nike Pro are meant to be worn as underwear with shorts or pants over top. They are a lot thinner (sometimes see through!) and not as durable. Though, I don't know how many track meets I've been to and their school issued Nike Pro tights as part of their uniform (mostly the sprinters). It is hilarious to see all these guys and girls running around in underwear!!
Running tights on the other hand are thicker, more durable, and are meant to be worn like pants with underwear, just like you'd throw on a pair of those tight soccer pants that the newer generation are addicted to lately. In fact, I can't even tell the different between soccer pants and tights no more, especially how tight these folks wear them. I even read somewhere that those soccer pants are a mens form of tights.
The easiest way you can tell underwear versus running tights - Compression underwear/baselayer V or U shaped front and back. Running tights will usually have a straight stitch in the front and the back. Some of the new Nike stuff has a U shaped (for the guys parts) and a straight hem for the butt. Personally, I like the straight stitch up front.
Here's what I'm talking about:
http://www.compressiondesign.com/two-types-of-compression-shorts-underwear-or-outerwear/
Anyhoo, I wear my running tights to the store, mall, etc. I've gotten a few flattering compliments, but a majority people really don't seem to care...