A working cocker or a proper Jack Russell would both be a different ball game compared to a standard poodle. They are much more intense, and often need an outlet for very strong working drives. People sometimes say that poodles are a working breed, but working line poodles are like hen's teeth, at least in the UK. I know a LOT of working gundogs (in the dozens) and exactly half a one is poodle.
There is a constant forlorn parade across this board of OPs tearing their hair out because their working cocker (which isn't work, nor given gundog training) is reactive/ resource guards/ never settles/ is anxious/ is a bloody nightmare. Not all WCS kept as pets end up like this, obviously, but a fair number obviously do. So be prepared to spend a lot of time keeping any WCS mentally active and engaged. And controlling its prey drive - which can be very strong.
JRTs can be little shits. I say this with love, as I have lived with several, including our own. Great dogs in the right hands, but bolshy, don't take no for an answer, will cheerfully take on a German shepherd and very prey driven (to the point of getting stuck down rabbit warrens). They often aren't all that interested in other dogs, though quite capable of living happily in multi-dog households.
You'd probably be better off with the cocker, but be prepared to either spend time finding one with chilled-out parents, or go for a show line, or have a gundog trainer on speed dial.
For context, I have working-line gundogs (I work one, and train both to a decent standard). They are bloody hard work - something I am entirely used to, but which I really notice when I have a walk that includes a local standard poodle - lovely dog, chilled, polite, kind, hangs around near the people, and so much less hassle than my two loons.
Edited TWICE! I need to proofread better!