Do I have Direct Mount or Hanger Mount? I'm very very new to bicycles and obtained a little project for me to work on, hence the amount of rust on the parts currently but I'm not sure if I have a direct or hanger mount derailleur? I'm 100% I need Shimano I just can't figure out which Mount haha.

I have attached a picture to the post (don't mind the disgraceful amount of rust and whatnot). Any help is appreciated!
That is a Direct Mount derailleur.  The hanger there is a separate part that is part of the frame.  When companies call them Hanger Mount or hanger style, the hanger is built into the derailleur and not removable.  To replace it, order the Direct Mount one.