Sure. I admit that even I generally find myself thinking of eastern martial arts when I think of martial arts. I agree it definately stems from the early days when the martial arts were first introduced to the west from the east. It was before my time so I cannot speak from first hand accounts but can see the point you raise.
Interestingly enough, the asian martial arts were only seen as "novel" after many European martial arts died out. Jujutsu seemed novel when there were no longer any masters of German Kampfringen. Kenjutsu is only exotic if you haven't practiced Italian Longsword. By the time Kenjutsu was brought to the West, the longsword hadn't been practiced in Europe for hundreds of years.
I am actually interested in studying some western martial arts as that is an area I know very little about. I hesitate to even name a few as examples as in the classical definition which we have been discussing, I am not sure if what I offered as an example would even be relavent...
Is fencing a martial art? Pankration? Wrestling? These are all western systems of one on one combat but they seem to be more martial sports... Will look into it further. So much to learn...
The difference between martial art and martial sport has always been a hazy one in Europe. Sporting systems were trained as an adjunct to training for lethal combat.
Sigmund Ringeck's fechtbuch says the following:
"Princes and Lords learn to survive with this art, in earnest and in play"
Which makes sense. It's hard to train the more damaging aspects of the Art safely. So the Europeans practiced both sporting and earnest martial arts. Jousting is a martial sport, but it sure helps you in using a lance for real on the battlefield. Hence it's a martial art too. The two things are not mutually exclusive.
Even today the same holds true. I have basically no chance of having to use a longsword for real. I will never have to face a sharp blade in earnest combat as I do not have a duel to the death planned any time soon.

But studying German longsword is most certainly a martial art. Besides, for most of us, the arts we study (no matter how "lethal" they're designed to be) are functionally sports since almost none of us fight to the death with any regularity.
Anyway Altc, there are all kinds of western combat arts that are not primarily sporting in outlook, even if they make use of sportive training (sparring, etc). Many are reconstructed arts from period manuscripts (Kunst des Fechtens, Armizare, etc) and other have living lineages (Military Sabre, Classical Italian Fencing, Canne d'Arme etc). The sporting arts are still very worth practicing and develop combative attributes. Give a Jogo Do Pau or Grande Baton practicioner a pool cue and he'll be able to defend himself well enough.
Best regards,
-Mark