There are different cuts of corned beef. The point cut seems to be pretty fatty and of course that is the one that is always on sale. There is an eye of round corned beef that is very nice and lean.
I cook my corned beef for about 3 hours on simmer. Unless it is an unusually large piece, I found it gets done in that time frame. I usually cook my vegetables separately because I am a "dunce" when it comes to cooking them together. It seems I always overcook the corned beef and it falls apart and is dry.
Everyone's tastes are different, but I buy the cuts with more fat, even the really fatty point/thick cut. I never buy the round as I find it to be too dry and chewy. I guess the best thing to do is experiment and buy two smaller pieces of different cuts and cook them together and see which one you like better. If you are really turned off by fat, the round would probably be best for you, even though it is definately not my own personal choice.
Whatever you decide on, it tastes great and I would make it much more often if my family liked it as much as I do. Enjoy your corned beef!
You must be on the west coast because when you typed that it was 6:37 my time...unless you're like my mom and say dinner will be done by 5:30 and it's really done by 7pm. ha ha.