The biggest issue is that an island will block your traffic paths for prep. We call that a "barrier island". It doesn't add to a kitchen, it makes activity worse. Draw a line from the sink to the fridge to the stove. Anything in the middle of that blocks your work. All of your work should be in a triangle one one side of the island.

Can the fridge be relocated on one of the legs of the L? Maybe where I see that tall cabinet on the left on the range leg? If not, then only a small movable type cart would be suitable, and even that will get in your way.

We actually relocated the refrigerator on that wall. We initially had a U-shaped kitchen and opened it to the dining room. In doing so, we had to find a place to put the refrigerator - on the other side of the cabinet to the left was not an option because that is an opening to the living room. I still want the island regardless if it blocks the triangle - I need the additional counter space and seating. Thanks for your suggestion though!

"best to have them backing up to the dining room table (people sitting at the island will have their backs to those at the table)?"

Yes, this.

If I were you, I'd make myself a temporary island of about the size you think you want. Make the "island" out of a table or some cardboard boxes or whatever. Set it where you think you want it. Place stools (or boxes) at the counter. Work around it for a few days. Pretend you're using it. You'll know pretty quickly what will work and what won't.

Why not build a mobile island? You can add locking casters and have stability and movement.

That way you can position the island for which position is optimum for the use.