What Hong Kong fish can be skinned, and can't be?

My doctor advises me, a woman, to avoid fish skin. As I don't speak Chinese, I’m linking to a restaurant menu to avoid any translation mishaps. Here are the exact fish I'm talking about with their English translations —

老鼠斑、東星斑、紅瓜子斑、三刀、海紅斑
Pacific Garoupa, Spotted Garoupa, Red Flag Garoupa, Morwong, Red Garoupa

At some restaurants in Hong Kong, staff automatically skin some of these steamed fish for you, without you asking.

At other restaurants, staff automatically cut up the fish, but don't remove the skin. When I ask staff in English to remove the skin, they freeze like rabbit caught in the spotlights, before apologizing copiously and explaining they can't physically remove the skin.

Why? Which species of fish can staff unskin, and can't?

As a picture is a worth thousands of words, here's the kind of fish I am referring to —

https://www.openrice.com/zh/hongkong/p-%E5%88%A9%E8%8B%91%E9%85%92%E5%AE%B6-%E5%BE%B7%E7%A6%8F%E5%BB%A3%E5%A0%B41%E6%9C%9F-p24288133

https://www.openrice.com/zh/hongkong/p-%E9%BE%8D%E9%96%80%E6%B5%B7%E9%AE%AE%E9%85%92%E5%AE%B6-p11214516

https://blog.carjaswong.com/2012/05/22/%E9%A6%99%E6%B8%AF%E7%BE%8E%E9%A3%9F-%E6%B5%B7%E9%AE%AE%E5%B7%A1%E7%A6%AE%E6%B5%B7%E9%AE%AE%E5%B3%B6%E9%85%92%E5%AE%B6/

I'm Chinese but I find this question very hard to answer. Mostly because restaurant service is mostly a lost art these days, so it's difficult to say what a server can or can not do. E.g. I heard in the good old days a server can de-bone a fish for you, i.e. butcher the fish, remove the bones, then put the fish back together. But for all my life I pick the bones myself when I eat and never had such service.

What I know for sure is a chef never partially skin a fish, especially one of such delicate kind. Although you can request the fish to be skinned before cooking, the traditional qualification of a chef's skill requires him to keep the skin intact. So the partially skinned fish of your first picture is more of sign of a mishap. This is normal because this fish is very delicate and tender after cooking.

Lastly, if you request the fish to be skinned before cooking, it should be OK. The skin of this fish is very thin and difficult to remove. I would imagine the chef would fillet the fish first, then skin it, then put the fish back together, then steam it. This will be very nontraditional and uncommon for steamed fish (common for fried fish), but not very complex or time-consuming either, so I believe most chefs are willing to cater to your special request.