alyssum or clover. Both come in different colors. Spreads just enough for the season...

Check your zone to see if it applies :)

The plant in the first picture is Mazus reptans, I believe. It requires constantly moist soil so probably is not a good choice for you. Something else green that withstands all day sun, is drought resistant and has pink/white/purple flowers is Thymus serpyllum and its many cultivars. It's hardy, perennial and can take a certain amount of foot traffic. I'm assuming your season is 12 months long and Thyme will give you leaves and/or flowers for the entire year. Allysum is an annual, several inches tall and would be squashed by walking. 'Clover' could be a lot of things.

The second plant I believe may be Pratia pedunculata. Also requires moisture and doesn't enjoy hot sun.

In the third photo the blue flowers are forget-me-nots. They are essentially a temporary spring plant. They look great for a few weeks and then get very ratty. At which point they'll need pulling up and composting. However, they will drop multitudes of seed which will give you the same look the next spring. They also need a reasonable amount of moisture and I doubt they'll be happy in full Ca sun all day.

In your situation, one of the problems with this kind of ground cover is that it works precisely because of its creeping nature. So hoping that it will stop at the edge of the lawn is a bit unrealistic.

The top two are Thymes so they should be OK. I do not know the others personally so I can't comment. If you ask about them specifically on the Perennials ForumI am sure there will be people with direct experience of all of them.

I would caution against Pratia pedunculata (Blue star creeper). It has been known to be highly invasive and difficult to remove as the stems and roots are extremely brittle, and any pieces left behind will regenerate. Have a look at some of the comments about it on Dave's Garden. I agree that any of the lower growing thymes would be good options. I have some elfin thyme planted between the flagstone on my patio and it does well with very little soil and moisture.

I have used Elfin thyme which is very slow growing for me. Also use wooly thyme which I love as it is very flat though I don't get much bloom. Years ago I sowed red creeping thyme and use it extensively as an edger for my paths. After first bloom it gets cut back short and will regrow and rebloom. Also find it easy to pull up if it spreads too much.

In one area a matt-forming low dianthus has wonderful tiny red blooms in the spring.

I walk on all except for the wooly thyme which seems more fragile to me.

Try searching for "stepables" for other suggestions.

Keep in mind that even those plants in the photos are only in bloom for part of the season.

Grape hyacinth blooms in one bed and dies back before the thyme gets much growth. Think Scilla or other tiny bulbs would work with thyme as well.

Regarding the Grape Hyacinth suggestion. It might die back in z4 but in my region and presumably the OP's zone it is evergreen and a rampant spreader leaving great lolloping clumps of scraggly foliage all year round. It would completely out grow and smother thyme, even culinary thyme, in my conditions.

Using an edging (steel or plastic, buried so less than 1/2" or so is above the soil and so the mower won't hit it) between the lawn grass and your ground cover will help to keep both from wandering. You may notice that your inspiration photos don't go through lawn - there's a reason for that since the grass will grow into the stones and the groundcover will grow into the grass. Alternatively, just use grass between the stepping stones and bury them to soil level so you can mow over them. Having the stones will help protect the grass, so you shouldn't end up with bare areas as you may have now. Be aware that bees will like your flowers, so if folks in your household like walking barefoot they will need to watch where they put their feet, and if there is someone who is bee-phobic, using flowering plants is not a good idea.

In my yard, the thymes planted along the walkway edge creep onto the stones. I currently have wooly thyme, but any creeping thyme will do that. IME wooly thyme grows lower and flowers for a shorter time than many others.

I have the Mazus planted between my stone walk way. I love it, but it does show up in the yard in a few places.