Depending on the amount of traffic, ajuga should do for you. It will spread, but is not a vine. stays close to the soil except when blooming. Herman

My neighbors have ajuga, and it is a bit of a thug. I read somewhere that any plant with "weed" in the common name should be viewed as a potential thug!

Not nearly as bad as English Ivy. Don't let Ajuga go to seed and trim the edges. Any ground cover spreads. It is not which one is a weed, they all can be. Just which one is easiest to control.

sweet woodruff. It can handle light traffic, and although it spreads very fast in spring, it is very shallow rooted and easy to remove. Far better behaved than periwinkle and will not choke out any other plants you decide to put in there. It doesn't like full shade, but will do fine in part shade or open shade.

Whatever you do, do not let anyone talk you into pachysandra.

I have Ligularia "Othello" on the north side of my house. Hardly any direct sun, yet it is blooming right now. Has been there a few years and not become a nuisance. It is evergreen after flowering. The roofer put his ladder in the bed and didn't even phase it except to bend the flower stalks

Yellow archangel (lamiastrum galeobdolon "florentinum") grows in my yard in shade, sun, wet,and dry places. It can come up where you don't want it, but mowing keeps it at bay.

Stringy sedum, grows right under the hosta, and pulls out easily by the handful (but never completly...so beware).
Golden Moneywort (or creeping Jenny) grows in sun and shade in my garden. It gets mowed over, run over, the dog pees on it, and it is STILL THERE. Its under hosta, in the lawn, under the ferns, its impossible to kill, but it looks nice.
Bishops weed (agopodium?> ten years of consistant spring pulling, and I still have patches of it under the maple trees I found while mulching the big hostas. I can't imagine why it is sold in stores, it should be growing rampant in ditches.

Sweet woodruff is considered an invasive species in some eastern states, because it has escaped backyard gardens and is taking over the state parks woodlands.