To be clear BA's allowance is one carry-on bag and one personal item, same as EI with the carry-on option. The only difference is not having the 10kg limit. They list a 23kg weight but for all practical purposes there's no weight limit -- I don't think they even have the facility to weigh bags at the gate and I've never heard of them weighing carry-ons at check-in. For all practical purposes the carry-ons are size limited and limited to "what you can lift into the overhead bin" (or get a friendly nearby passenger to lift for you). You would be pretty hard pressed to get 23kg into a personal item short of carrying gold or lead bricks though. It's not even all that easy to get 23kg into a full size carry-on rollaboard.

Aer Lingus does have a 10kg limit so check-in staff might decide to weigh carry-on. The only time I've seen it is at an outstation where contract staff felt they had to follow the instructions to the letter. It was hilarious seeing them insist on weighing a tiny 13l backpack that weighed maybe 1kg. That was years ago though. I'm not really sure what the point of weighing carry-ons is anyways -- you can always remove jackets and wear them or put heavy objects in your pockets until you're past the check-in desk and then return them to your bag afterwards....

Fwiw if you want to be sure when you check in, request an e-mail boarding pass (you can get a mobile app boarding pass or SMS boarding pass too). The e-mail boarding pass will clearly say "CARRY-ON BAG" on it if your ticket includes carry-on. That means you get the 10kg carry-on in addition to the personal item (which you always get).