The way we used to eat corned beef and cabbage when I (Nate) was growing up was to take one of those cryovac'd corned beef briskets, cube them up into chunks, and toss them into a pressure cooker along with carrots, potatoes, and the spice packet. Then we'd cut up a head of cabbage and boil the life out of it in a separate pot.
The end result was very tender brisket, but the veggies were falling apart and the cabbage was tasteless. We'd eat all of that with cheap French's yellow mustard.
In Annie's house, "corned beef" meant the hash that came out of a can. Her mum would fry it up with egg and serve it on bread.
It wasn't until Annie came to Hawaii to study that she found out what "real" corned beef was. Her host family served it to her out of a slow cooker, cut into thick slices. Then she introduced that style to me.
Now, our local Lunardi's grocery store sells fresh corned beef. Annie bought a 7-pound brisket point and flat cut. She simmered it for several hours, adding the potatoes and carrots at the last hour. We boiled the cabbage until they were just cooked, still retaining the green color.
I like the fresh corned beef because it is less salty than the cryovac'd ones. The veggies arent' mushy but retain their texture and flavor.
Someday I may attempt to corn my own beef brisket.