I've always described myself as "vaguely ethnic" and I find the term POC downright offensive. I don't understand why it's suddenly become acceptable. POC is just a slightly different way of saying "coloured" and means "I have no idea what you are, but you're not white".
I prefer "vaguely ethnic" as people often can't place my ethnic background due to sheer ignorance, and it doesn't give them any clues and puts them deservedly on the back foot. I'm fed up of the "but where are you from?" questions. England, that's where!
People can stop telling me which terms I can and cannot use, thanks. Actually having the genes that I do allows me to choose whichever words make me feel comfortable.
Onto other pointless words... in this country, "Asian" only means "Indian". In the US, "Asian" can mean Chinese. I've given up trying to reclaim the word. It's never going to work in the UK.
I don't find "I'm having a Chinese" offensive because people clearly mean "a Chinese takeaway". If they wanted to be offensive, there are other words to substitute for a Chinese takeaway that racists can and do use. Intention is everything. When you directly experience racism, you know when people are discriminating against you, and when they're not.
I think saying a Chinese or an Indian is more common because people typically order several courses, and it fits their definition of what Chinese or Indian food is. It's not well understood, because it's complicated to make. I think there's an understanding that most pizza places do not serve authentic Italian pizza, so that kind of takeaway is referred to as "a pizza" rather than "an Italian". Other "Italian in the UK" food either doesn't work as well as a takeaway dish (pasta) or is on the pricey side (meat/fish course), so a term doesn't come up as often.
The fact that people abbreviate to a Chinese or an Indian just shows those two cuisines are the most popular for takeaways. FWIW, I think people are more likely to dine in with European cuisine.