Too blurry to call decisively.
Looks like some solder mask is scraped off in both cases. For the first picture (Pic A in the page title) it could also be that the trace is broken, or possibly shorted to the other parts.
Solder mask is a non conductive layer applied on top of some circuit boards to prevent conduction, note what certain things are supposed to be and possibly help with corrosion issues as well. Obviously you can't solder to it and expect much to happen either.
Boards are made by putting layers of copper on layers of silicon carbide or something. The chemicals or tools to in turn remove the copper are expensive and time consuming so typically only what is needed to be removed to make it work is removed, as opposed to removing everything that does not serve a purpose. Usually large sections on a board are used as a ground or tied to it, but they have to be.
I don't know what method offhand that the 360 sticks use (some things will detect when something has a voltage going in, some things will detect when a given put is grounded out, sticks use a few more complicated methods but still based on that) but a broken trace or shorted trace could easily lead to it being detected as pressed all the time and would be what I look at first.
If it is shorted then sort that out. It will usually be because a blob of solder did something or because a trace was lifted and carried across (see below in that case as you will probably also have it broken somewhere along the line).
If it is cut then you will need to fix it. Fixing a trace in such a small area is not an impossible task but a fairly hard one and if you are learning you might want to practice on something else first. Alternatively if the trace runs from one point to another then you can bypass it with a wire -- far easier to solder from a leg of a chip to a leg of another chip, or hole to a hole than scrape off the mask and solder to that. There are also quite expensive conductive inks (pen form and brush forms are available) that are tricky to use and I don't like using them for a board like a controller where it is likely to take a bit of flex and abuse but do work. If you are using ink then make sure the solder mask is removed from the relevant spots, and also that it does not go across to other areas of the board and short something out.
Also a few things there look like lifted pads. This usually happens because of too much heat (if you only have a small section of copper remaining for it then it only has that little bit of surface area of glue holding it to the board), too much force removing the item or a combo of the two. If there is no pad then there is not going to be any conduction. If the pad is still there some get lucky gluing them back down with epoxy or superglue (remember superglue evaporates when it gets hot and you don't really want that in your eyes if you are planning to solder it again afterwards, though do it right and you can still solder it). If it is not then you get to do the same fixes as before and run a wire to a trace or its ultimate destination to restore the ability of the thing to conduct and do what it needs to do.
Similarly did you dig in and slip with the soldering iron in that? If you are doing electrical soldering and applying enough force that if you slip you end up gouging out a chunk of board you are doing it wrong. Modern devices often have lead free solder which is not as good as good old glorious leaded solder so if you are removing things you are advised to flow a bit of leaded solder into the existing solder connections first to make it easier in the long run.