The yellow green foliaged plant is one of the low growing Euonymus such as Euonymus japonicus Aureo-marginatua or perhaps a Euonymus fortunii cultivar. This question would be better placed on the Name that Plant forum for ID of your other plant.

Have you tried just pulling the other plant once a week for a season?

Thanks very much for your responses. Looks like we have some digging to do. Going to also put some lime down in an effort to discourage the creeping buttercup, and possibly some Preen for good measure. Preen's going on the lawn soon anyway to discourage weeds in general, so what the heck. Thanks again.

The weed-killers that are not general all purpose (like Round-up) usually separate broadleafs from grasses or grasses from broadleafs. There are none that I know of that separate certain broadleafs from other broadleafs (what you have here.) When they are as mixed as your picture shows, usually, I would dig the whole bed, including all root pieces, separate out the bad plant and replant the good ... basically, starting over. But it's effective and usually the good plant recovers rapidly. Sometimes, it's impossible to remove all the bad weed root pieces and then the whole bed must be killed and started over.

There are some plants that Round-up (or Weed-B-Gone) doesn't kill and once in a while, it happens to be the plant you are trying to keep. If that's the case, it might be able to work to separate out the undesirable plant. However, this is totally up to experimentation and there is no list of plants that it doesn't kill, so it's a method of last resort when you're prepared to kill everything and start over.

Pulling weekly as Babs suggests is labor intensive but offers promise.

Thank you, Yardvaark. Good information.

It's not such a big job. Get yourself comfortable sitting, kneeling or crouching on the grass and tackle it with a hand fork. You need to get under the central rosette and remove all the roots. You'll find you start to zone out and it's actually quite relaxing. If the ground is compacted water it well before starting. Might take forty minutes, tops.

I have come to enjoy weeding more than any other garden task. As Floral says, get comfortable! For me, it's early morning, when the ground is soft and the light make everything including me look better. I enjoy the tweeting of real birds:) Second best is evening, when bugs are ameliorated by a glass of wine. Why is weeding so pleasurable? It's like editing, where extracting a single word can improve the whole essay. The trick is to get close to the ground. Sit, don't bend. Feel the soil. Appreciate the diversity of plants that you miss from five feet. And when you're done, no longer how long you've been at it, the effect is immediately gratifying. Unlike new planting, where you step back and your puny additions are lost in a sea of mulch, here a short time yields a finished vignette. Now I also appreciate the pleasure of the macro end of the gardening spectrum, riding a powerful mower. In the time I may complete a bed, my husband has turned acres of weedy, lumpy lawn into a verdant carpet. It's all the in-between tasks that are tiresome.

Everything kitasei says I second. You learn a lot from weeding including identifying plants at their very early seedling stages.

And if necessary wear your reading glasses, or as I prefer to say, my weeding glasses.