It appears you're looking at Spring Hill's collection.

If you enter the common name in their search function, and go to the individual listings, it confirms your findings on their Latin names.

Diamond Grass is the only one I haven't heard of, and Google finds it used for both Calamagrostis brachytricha and Achnatherum brachytricha. Bluestem says both of those are Stipa brachytricha. According to Greenlee, the correct name for Calamagrostis brachytricha is Calamagrostis arundinacea var. brachytricha.

Confusing, isn't it?

I searched & found all but Purple Love Grass and Strawberries & Cream Grass on the site, but these grasses are a bit easier to id. Yes, I had a hard time with the Diamond Grass. There seems to be some mass confusion on the latin name. I have found that Strawberries & Cream is a danger to wetlands as it's so invasive in that situation .. I'm not sure about the diamond grass (korean feather grass).

I can't find any reference to Calamagrostis brachytricha being invasive, but I know it can be grown from seed, because I started a few clumps of it last year. They didn't bloom in their first fall, so I can't say if they'll develop viable seed in our zone. I expect they will.

I checked several well known resources on ornamental grasses on my bookshelf and found that all of them mention Calamagrostis brachytricha as the proper Latin name for diamond grass. Quite a few give Achnatherum brachytricha as a synonym and only one mentions Stipa brachytricha as well.
For the rest, I have to agree with Donn. I've never heard of Calamagrostis (or, for that matter, Achnatherum or Stipa) brachytricha being invasive. I wouldn't mind if it were invasive either, because it's a superb plant and I'd welcome it everywhere in my garden. For that reason, it's a bit of a disappointment that I started a few clumps from seed myself two years ago and I've not seen them blooming yet. I'll keep my fingers crossed for this year.

The entire genus of Stipa has been reclassified, so that may account for some of the taxonomic confusion. In many areas, Phalaris arundinacea is considerd a very aggressive spreader and it doesn't care if the soil is moist or not - it is on my state's noxious weed list.

I have actually purchased this collection. The one grass that I have totally crossed out is Phalaris arundinacea as it is considered invasive. I have gathered the other grasses have potential to spread but are not nearly as invasive. I hope to be corrected if I am in error.

Hard to believe such a whimpy grass (Imperata cylindrica 'Red Baron') could be invasive. It's not that I don't believe the reports, it's just that in my garden the darn thing barely survives.
Another case of zone envy! sigh .....
A.

You're exactly right about Imperata Red Baron.

I grew this for years before it made the weed lists. It really got bulked in as a generic cogon grass, but the variety is not a fast spreader by any means. I have, however, seen what the reverted ones do and they can really spread.

I should mention that I produced this from divisions and NEVER saw a revert, but those grown from tissue cultured plants did have some reverted...hence the baby gets thrown out with the bath water.