Using your phone in a meeting

Taking notes on phone is absolutely fine. It's done at plenty of buyside and sell-side meetings. Even with management. And I'm in London. Sometimes there simply isn't room for a laptop/iPad or it isn't particularly convenient. 

Taking notes on a phone is pretty common now, but it can still come across as rude because people can’t tell whether you’re taking notes or just scrolling. In a formal meeting, I’d probably stick to a notebook or laptop to avoid giving the wrong impression.

I've never seen a very good investor taking notes on his phone

You can bet when the CEO of a company meets with his own people, no one is taking notes on their phone (nor wearing Patagonia vests). It looks terrible. This is typical HF investors being completely unaware.

It’s public markets and you need to be responsive during trading hours. PM is sitting in a meeting with someone for 45mins and will be receiving trade emails & position correspondence from analysts. These need to be reviewed immediately in case something is time sensitive. Anyone who interprets that as rude, generally doesn’t know how HF public markets work. Thats why my firm limits the amount of time PMs have to be meetings to shorter intervals. So they aren’t away from the book too long.

It's completely fine as long as you demonstrate you are clearly engaged and participating in the meeting. Being in the corner typing on your phone and never saying anything will give a poor impression to some management teams.