If I understand your story and pictures correctly, you're asking for ID of the grass that you have at the top of the hill that is doing a good job of keep weeds at bay ...? Probably Liriope spicata. You can use existing to make starts. It is invasive, which is why it's doing a good job on the hill. Figure out a way to stop its invasion at the bottom of the hill. Some kind of root barrier and vigilance is in order.

how deep of a root barrier would I need to have? What are cheap and easy options for such a barrier? Will this kind of liriope just start growing into the lawn at the bottom of the hill?

what other ground covers could I use that are not so invasive as to where they would start invading my lawn below? I need something that crowds out weeds, and could add stability to the soil. Something that would also look good planted below the existing levels of Liriope. Thanks.

To answer the question of how deep the root barrier, dig a hole next to the Liriope and see how deep the roots extend. Not all parts of the roots spread sideways. See if you can determine the "danger zone" of the root parts that spread. The root barrier should be somewhat deeper than that. I think your best bet is to stick with one type of groundcover. Two mixing generally does not look good.

Here are some shots of the hill I just took from the bottom looking up. I got some different angles. The Liriope is at the top 1/3 of the hill....and I discovered a smaller patch of it towards the bottom. There are some dense patches of some sort of ivy below the Liriope. The middle section of the hill was covered with tall weeds/grass which I ripped up the other day. So I have some bare patches to fill in. I thought maybe some Drift Roses I've been reading about, or creeping phlox? Something to give the hill some additional color.
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Unless you first eradicate weeds and keep them out while the "good" plants grow in, the hillside will never be controllable or good looking. It's essential to obliterate the weeds. In addition to herbicide, it will require mulch and monitoring until groundcover grows. In theory you could dig weeds out, but on a hillside, that's asking for additional problems by loosening the soil and making its surface unstable. If you want to incorporate a groundcover that will add color, like creeping phlox, it would be better to section off a portion of the hill and devote it to that plant, rather than interspersing it among other groundcovers.

When you say there is a "hillside" and a "yard below". Can you explain the difference or offer more photos.

As far as the Liriope goes you are lucky to have it. You can divide it and get many plants for free.

Root barrier can be found online. I have used whatever I could find in my junk. Unlike you I did not need a barrier for a large area.
I love your roses but roses need pruning. I could never scale the slope to do that each year. If you are more agile then go ahead.

Even if the Liriope grows into the lawn, regular mowing of the lawn will inhibit its spread long term. An 18 to 24" deep barrier ought to be sufficient.

Even if the Liriope grows into the lawn, regular mowing of the lawn will inhibit its spread long term. An 18 to 24" deep barrier ought to be sufficient.

@emmarene
If you look at the first sets of pics I posted taken from the top of the hill looking down, my yard starts to flatten out. you can see my neighbors wood fence slightly in the picture.

I keep thinking of adding some more color to the hill. In the spot on the hill where you can see the black patch of soil, I was thinking some Drift roses in that section.

bgaviator,
WOW. That is steep. I see you have stairs on the side
of this hill to access the back yard.
Lirope is wonderful for that hill, I don't know who planted the exsisting lirope, but they also planted grasses up by the house, which is also terrific for errosion.
I would wait til spring and then buy some more of that lirope.
Just so you know, there are 2 kinds. Running lirope, which is also called Monkey Grass, which doesn't get quite as tall as yours, then there is the clumping kind.
The clumping kind doesn't spread like the running kind does. It gets fuller and longer, I think that is what you have, I could be wrong, but look for the clumping kind if you don't want anything spreading into the grass below.
You really need something on that hill though.

yes @butterfly4u, it is steep. It made us very hesitant about buying the house, but we liked it overall too much. You should see on the other side below our deck. We plan on putting a multi-tiered retaining wall in at some point. The grass on the hill is downright dangerous to mow!

If grass is dangerous to mow, it's a maintenance problem and should be replaced with something that doesn't need mowing--low shrubs of many types come to mind--junipers, for example.

Would Asian Jasmine be good for this hill possibly? I've read its really good for erosion control too. I just worry about invasiveness with any of these ground covers