This is a good example of how bilingual dictionaries can become pretty useless at times. They just give you the counterparts in the other language; They do not define the words for you.
「根気」 is the will power that one needs to have in order to continue doing the same thing for an extended period of time. Close to "untiringness".
「忍耐」 means enduring one's distress, pain, anger, etc. It is all about one's mental strength against one's own mental hardship, not directly against something external.
There is some interchangeability between 「根気」 and 「忍耐」, better yet 「忍耐[力]{りょく}」, with the latter being the bigger (and less "everyday") word.
「我慢」 refers to enduring by supressing one's emotions and physiological desires. It is a somewhat colloquial and "everyday" kind of word. Parents use it often when talking to kids. Mostly used to talk about relatively short-term activities such as holding a pee, not eating between meals, etc.
「我慢」is closest to 「辛抱」 in meaning among the words in question here. It is farthest from 「根気」 and 「忍耐」.
「辛抱」, in my own words, is like the long-term version of 「我慢」. The two words are indeed often interchangeable and when interchangeable, 「辛抱」 would sound slightly more old-fashioned. It is the kind of word that you use less often than your parents do. I feel it safe to say that we use 「辛抱」 to talk about more abstract (and larger-scale) things like one's environment and living conditions.