backpacks or other bag for Italy sites

We were in Mary Major a few weeks ago with a small nylon backpack, no issue.

And fyi - the Borghese gallery has a great system for this. Picture a revolving hanging rack like at your dry cleaners. Put your stuff in a big bag they provide, they give you a tag number and hang this bag on the moving rack which runs under the front parking area. When you pick up, give em you ticket# and they scan it and the rack rotates until yours appears. Just like dry cleaners.

We were also at duomo in florence a few weeks ago. There is a bag check just inside on left. They take you to a room with various size lockers and you put your stuff in empty one and take the key. If you are with a big family or group there are bigger lockers to put everyone's stuff in one locker to conserve space.

Just needing to carry a water bottle, small guidebook and a jacket.

For those, you might consider a smaller cross-body bag, or even pockets. I have a light jacket with pockets large enough to carry a small guidebook and a bottle of water. There are several cross-body bags that have water bottle pockets, as well.

And Rick suggests wearing a small backpack slung over one shoulder or tucked under you arm, as ways to defeat the dreaded bag check.

The bag check is the best. If you have a jacket and a pack while in a museum it often becomes a bother to carry, and you get hot. It seems so much better to me to meander in the museum without all the baggage.
They tend to be pretty strict if there is a backpack policy. Otherwise people have all kinds of excuses. Go with the flow and don’t argue.

We were at the Duomo in Florence a few weeks ago and we saw people with backpacks. You can check them when you get in (edited - I don’t actually remember seeing bag storage for the Duomo. There was definitely storage in the museum). I carried a crossbody purse and it went through fine.
You will need to go through security at most if not all of the museums and sites.