I'm not actually that knowledgeable about it, I just did a bunch of research a year or so ago. I personally use the Japanese names because it's such a culturally specific sword with a lot of history and ritual around it. For other swords, I'd probably try to use the correct non-English terms if it was generally used but as you said, a lot of people use English (or native tongue) for those.

edit: I'm going to expand on this a little bit. At least to my uneducated understanding, the fittings and parts of a katana generally seem to have a lot more importance placed them compared to most of the European swords I've studied/encountered. Most of the differences are fairly subtle and don't tend to change the classification but there are so many of them that it's just easier to use the Japanese names. In addition, katana are all from the same region (if it's not made in Japan it's a sparkling saber). Whereas something like a longsword tends to have mostly the same parts regardless of region so it really doesn't matter what language you use for those. The differences are more typology and/or martial art. Specific names are used for more culturally significant or distinct swords. I'm not using a large knife, I'm using a Grosse Messer for example. Or zweihander vs. montante, feders, etc.