First of all, I have no idea what happened to this gutter in the first place (main picture). It looks like somebody hacked a section of gutter out with a dull steak knife. This is at an apartment building which is a pretty common place to find fix fails. Unfortunately, a lot of apartment maintenance staff are not experienced with proper residential repairs. It seems that a maintenance worker came out and smeared caulk all over the seams of the gutter. The problem is that just smearing caulk over the seam isn’t going to keep the water in (clearly, as illustrated in the picture). The best way to repair leaking gutter is to pull it apart, clean the area, then put a bead of caulk BETWEEN the two pieces of gutter you’re trying to seal. Then you can do some extra on the outside as well, but you need the get caulk between the pieces to really glue them together.
If you have the budget, getting old damaged gutters replaced with full pieces of extruded gutters is the best option. Any time there is a seem in your gutters, it will eventually leak.