Beef – use casserole steak instead of braising beef for this Chinese Beef Casserole recipe

Generically speaking meat that is appropriate for a braise is tougher and has connective tissue that can be turned to gelatin by the long slow cooking process. As you've noted, meat that is tender can be "cooked to death" using that same method, so I would, generally, recommend against using a braise.

However, a stove top braise can go quickly without ruining the meat, if you keep it short and treat the braising liquid as more of a sauce than anything else. I would suggest that you brown each side of your lamb chops, then add all your other ingredients. Depending on the amount of liquid your original recipe calls for, you might want to cut back. I wouldn't want more than 1/2 cup or so of liquid. "Braise" covered on the top of the stove for 1/2 hour, never going past a simmer. Pull out the chops and cover, while you reduce the braising liquid to make it more sauce-like, then spoon onto the chops.

Note that you won't have the long time to meld flavors, and if there are big chunks of garlic or onions, they won't be a sweet as in the longer braise. But you should get a serviceable dish.

Pot Roasting = Braising

This more recent document from the same association, Cattlemen's Beef Board and National Cattlemen's Beef, supports this by using them interchangeably: 3 Simple Steps For Braising/Pot Roasting Beef

I was trying to figure out what the document from your question may have been implying by the cuts the different cooking techniques were recommended for. And I'd guess, because Pot Roasting is applied to more connective heavy cuts, that braising would be a shorter cooking time. That's my best guess.