I'll give you my thoughts but I really urge you to get someone in who has specific experience in dealing with basements with water problems. Plenty of guys know how to build walls, etc. but really have little actual experience with the water and/or mold issues. With that disclaimer, here are my initial thoughts. This is going to be long, because I want to try and help - I've been in a similar situation as you and I know it is no fun, and information is hard to come by.
First priority is to determine and fix the CAUSE of the water problem, not just treat the symptoms by putting on epoxy or whatever on the cracks. As you've said, you've had several known leaks in just a couple of years. That's a problem. The usual culprit is some combination of exterior landscaping and gutter management. At this time of year, it sounds like landscaping. Snow melting/water runoff should be forced away from your foundation to the extent possible by proper landscaping. Unless and until you find and resolve the root cause, you will have to anticipate future water problems.
Secondly, I - personally - would definitely get rid of that existing insulation on the perimeter walls immediately, for three reasons: (1) you need to be able to clearly examine all your foundation walls to be certain you are aware of, and fix, all the cracks, and you can't be certain of doing that with the insulation still up; and (2) the existing insulation has likely already gotten wet somewhere, and once that happens that is a mold nest just waiting to grow; and (3) even without the water concerns, it is clear your builder did a poor job of installing the insulation, as is evidenced by the incomplete wall coverage and questionable direct wall attachment. Put all three issues together and it would be a compelling argument, in my opinion, to just rip out what you have. I certainly don't think the idea of a wire or something behind the insulation would be of any help mold wise (if you're curious do some research on how mold forms and grows and you'll learn, as I did, that once mold exists your only real certain cure is to completely remove any organic material which it has touched).
Once you got the old stuff out and had not only fixed the cracks/leaks but had also made certain you had discovered & addressed the source of the water, here's how I would go about insulating (although you could probably get several differing opinions on the method used - I know I certainly did!)
I would definitely use the polystyrene or "rigid board" insulation, applied directly against the walls. This both insulates and acts as a vapor barrier. This comes in varying thicknesses, with the thicker ones having higher R values and costing more. Installing this in your situation presents a few options.
Obviously you could disassemble your framing so you could get easy access to the walls. I don't know how you affixed your framing to the floor and ceiling so I'm not sure if that would be a huge problem or fairly easy. (At this point I'd ask another related question: it's hard to tell from your photos, but did you use treated lumber for you base plates? They don't look like the typical green treated lumber.)
Second, you could cut them into widths to fit inside the studs, but I wouldn't recommend that unless you had framed up against the wall, because it would just leave too many uninsulated gaps around the entire wall.
A third possibility - given that you apparently have about an inch or more between the framing and the walls - would be to selectively remove a few 2x4s sufficient to allow you to slide the sheets of insulation into the gap and then replace the 2x4. This might be a bit trick, as you'll have to make sure the height of the sheets is snug or at least close to the top plate, but it might just work. Then since they wouldn't be glued to the wall you can get a similar result by simply putting in shims of some sort (likely the scrap insulation cuts) between the framing and the insulation, forcing the insulation to remain snug against the wall.
(Edit: one other possibility worth considering, that I have no first hand experience with: the spray foam insulation against the walls, which you could do without fiddling with your framing. My understanding is this needs to be done by an expert, so the cost is likely higher.)
Once the polystyrene is in, you can additionally put in UNFACED fiberglass insulation between the studs if you want to increase the R value, but that isn't essential.
That's my 2 cents worth. I know just contemplating dealing with this issue at this stage of your build is very frustrating. I know that doing what I suggest means delaying your project and adding cost. If you decide to proceed anyway or without doing all of what I would do in your shoes, you would probably find others who would do the same thing. There are trade offs and risks either way. But to me the risk of not doing it now and hoping you won't have a problem in the future is high, and the cost and hassle of dealing with the problem later after you've done all the more expensive finishing work is far greater than the cost and hassle of biting the bullet and doing it now.
Whatever you choose to do, good luck.