I'm completely self taught, so I struggle through theory only when it interests me or is necessary. But lately I've begun to analyze why I like certain types of music, mainly baroque, and it occurs to me that counterpoint is probably the reason. I try to apply it to other genres and I realize I'm not 100% sure what it is.

By my own naive observations, I believe it is having multiple, independent melodies played simultaneously to produce harmony. The key words being independent melody, as in, each melody can rightly stand on its own without accompaniment. If a "melody" is not an independent line of music, then by definition it isn't counterpoint, even if it harmonizes.

So am I on the wrong path here, or am I pretty close? And by that definition, any style of music can make use of counterpoint regardless of genre, rhythm, etc yes?
You're on the right track. Although, the term itself has historically been used in different ways. Contrapunctus was a term used to describe organum, descant, and various other "note against note" musical pieces, regardless of line independence. In today's usage, however, "counterpoint" is generally used in reference to the academic study of polyphony.
^This. Counterpoint is multiple independent melodies. Or, polyphony.

And yes, counterpoint can be used anywhere.
"There are two styles of music. Good music and bad music." -Duke Ellington

"If you really think about it, the guitar is a pointless instrument." - Robert Fripp