Workout equipment at home will most likely gather more dust than what it should, so I’d recommend going to the gym.
You don’t need to spend a lot of time at the gym at all.
If you’re doing cardio, then 15 minutes with high intensity is good, then you can increase to 30 minutes on days you where have more time. Once you’ve started and if you manage to keep it up fairly regularly, you will soon notice that your health is improving and you’ll probably also want to spend more time there. Definitely 15-30 minutes after work, that’s my recommendation.
Some daysm, when I really don’t want to go to the gym, I go anyway and do a quick warm-up (5 minutes) and then concentrate on just one muscle group. For example I only do arm exercises, or only chest. Then I’m done in no time and a little bit of work out is way better than no work out.
Also it’s easier to keep it up when you “go somewhere” (the gym) to work out. At home there’s always a lot of distractions: computer, house chores, family visits, TV, phone, all kinds of things that will make you say to yourself: “I’ll do it later/I’ll do it tomorrow”.
I don’t know if you have this saying in English: “The door mile step is the hardest”. Well, the door mile step will definitely be hard if you don’t even have to exit the door.

I really like your thoughts about commitments rather than resolutions. This was very well written.