Adventures at the vet - was I rude?

Yes, pets are like children to many owners, but like with special needs children it is ultimately the parent's responsibility to make sure to discuss how to best approach them with the vet/doctor before proceeding. This is infinitely more important for difficult patients such as your cats. So the takeaway I get is that you'll need to have a more detailed discussion with your future vet how to specifically handle the cats.

I suppose there may have been something lost in translation as the instructions went from you to your boyfriend to the vet. Depending on how the conversation went, perhaps the vet had the idea that he was being advised to restrain the cat with the net so that he could be safely examined. Things only escalated from there.

Of course your behavior at the office could have been better, at the least it didn't help the situation. But it's easy to smooth over, and you could return to apologize and offer to pay the appointment fee. Or not, in the scheme of things it doesn't really matter for anything, but it might end up helping to improve your relationship with your bf since it's definitely one of those cultural understanding moments all too frequent in international relationships.

And in the future, if you can't explain the needs of your cat well enough in Japanese then it's best have a detailed talk with your bf in advance of your next appointment so you and him are on the same page going in.