I have both nerites and mystery snails. The nerite will not eat anything except algae (to the point where he starts wandering down the tubes of my sponge filters to get some). I'm not sure that they would climb up the grass to get at it though, mine prefers climbing more solid routes.
The mystery snails will nibble at the plants if there's more interesting food on and around them. My lily's leaves have some pretty heavy eating marks all over them. I love my mysteries and have enough plants that I don't mind them munching a bit on them.
Nerites are great for glass algae. Mystery snails are less picky. As far as I can tell all of the other types of snails for our tanks tend to reproduce tons. You could rectify that though by stocking MTS or whatever snails and when they bloom add an assassin snail. Assassin snails will munch on nerites or mysteries though so you have to make a choice of which you want more.
Also maybe consider ghost shrimp, red cherry shrimp, or other shrimp. They can be great algae cleaners too and will definitely enjoy eating it off grass.
My 20 gallon community: 2 Mystery Snails - Arnold Sluggenator & Goldie Brawn, 1 Zebra Nerite Snail - Poop (my pet name for my boyfriend), 1 African Dwarf Frog Juan - Frog, 9 Male Endler's Livebearers - Cloudy, Clyde, Conga, Darius, Frank, Lemon, O'Rien, Peephole, Urkel & Various live plants - anacharis, anubias, hornwort, java fern, java moss, narrow leaf chain sword, water lily, water onion. The Prize Jewel - Dragonscale Crown Tail Betta - Sylvester "Sly" Capone