Upon closer inspection I think they are the same plant - just don't know what it is?

It's catmint!

Glad I didn't pull it!

The second one is Motherwort- nasty pest in my yard.

Thank you both! I will better investigate the second plant, you are right they are not the same.

I've never planted any motherwort in my yard - is it easily carried from other yards?

It reseeds and naturalizes- used generations ago for medicinal purposes so and yes it's carried though a burr like seed pod that sticks to fur and clothing and it's sharp so the darn things hurt.

Confirm motherwort, which is a mint, by examining the stem. It should be square, which is a dead giveaway that it is a mint. If the stem is not square then motherwort is probably not correct. Then I would say it sorta looks like the Canada anemones I used to grow. They moved around all over the yard, too, and eventually moved out.

Yes, it has a square stem. I've potted it up to plant elsewhere to see if I like it - will it serve any purpose to plant in a bottomless container to prevent spreading? If not, how fast does it spread?

Typical mints are like 'give them an inch and they will take your yard'. Most of them send out lots of stolons and root out where ever a stem touches the ground. Those that reseed themselves like this one tend to not only do so very aggressively, but also have seed that can lay in the ground sometimes for years. Confine it and do not let it bloom are your best bet if you are going to try it as a medicinal herb. Confine it to a 5 gal pail with holes punched in the bottom for drainage and don't let the stems hang over the edge of the pail. The flowers are interesting but there are prettier ones that are not as invasive, if you are looking for an ornamental.

FWIW most mint teas will be relaxing if that caught your eye. You can also use Monarda for that kind of tea as well as ornament for a tamer and much prettier mint.