Is there any other religion, other than buddhism, that claims a connection/co-existence/merge (whichever is the proper term) with Zen philosophy? sure there are individuals either thinking they are, or are essentially in spirit doing so...but is there anyone making the claim of "Zen Confucianism" for example?
If not, then why the distiction by combining terms 'Zen + Buddhism' ? The reason is in the popularity of the combination, such that the combo 'required' definition. It's just a label to describe what a bunch of people believe/practice. and I suppose like any other philosophy/reality it often isn't sought...it finds you....and I'd also suppose that if it did, the label would no longer have meaning.
As far as a named philosophy and martial arts, I see many more connections to Dao. The philosophies in Dao relate (if only in metaphor) to physical philosophies of combat. They seem to be better suited to strategical/method philosophy. Whereas Zen philosophies seem to relate more to the state of mind during such practice, preparation and/or AFTER such practice. If you are thinking Zen DURING the act, then you missed it's point.
It's almost like a conditioning of the mind before and after. whereas Dao/Tao addresses wisdom WITHIN the action.
practicle example: you are sparing in class against an opponent you know has more skill...you are bunched up, stiff and nervous. you take a moment between 'rounds' ; close your eyes, tell yourself to breath easy, tell yourself to relax, tell yourself to think no thoughts...you fancy yourself as incorporating Zen philosophy 'mushin', or another word you learned/were taught. It's not Zen. What you just did was a relaxation technique using visualization and self-suggestion. which is good stuff, it works. but from what I understand about Zen is that it just is a state of mind. To train for that 'just is' state is subject to debate, and I certainly don't have the answer. It's a non-way that everyone finds for themselves I suppose.
another example: 'the last samurai' movie. when Cruise has that practice drill with his instructor and then end in a draw. just before that it shows Cruise relaxing, visualizing, then 'letting go'. enabling him to allow his head to process more information in real-time due to the fact it's not bogging itself down with details. It has the effect of time slowing down since your brain is able to keep up and surpass the moment. ...it's not Zen.
'Zen' is just one type of elusive training to be ABLE to enter the stream....once there it's non-Zen.
The reason Zen and Martial Arts are not inseparably linked is because there are other ways (for the purposes of MA) to get the same effect....and many DO choose other ways. hours upon years of physically practicing what you wish to be able to do without thought is another way, 'second nature', 'in the zone', etc. people who do that and never meditate or speak in terms of 'Zen', are still able to enter the non-Zen state. Therefore your philosophy and whatever MA you practice is your own link and does not require a term....but put it there if you like...just don't assume it's the 'right' way or only way or best way...it's just A way - and it's certainly not tied to MA by definition. from where I'm standing today anyway - I'll probably evolve how I look at that.