My guess - on what was meant by the boxing vs. martial arts thing - is that typically boxers are trying to knock each other out, and marital arts practitioners are trying to score points.
Obviously there are lots of exceptions but, a lot of "black belts" never progress beyond points-sparing and preforming katas.
I think it's a good idea to get some hand-to-hands training of some sort, but for practical purposes, the discipline you choose probably doesn't matter much.
If you're comfortable enough with physical interactions that you won't freeze up, have a basic understanding of kinesthetics, and can avoid having your gun taken from you, you'll probably do pretty well at defending yourself.
si vis pacem para bellum