Neither. Both are inefficient. Give us a 2D floor plan with all measurements and also show how the kitchen relates to the rest of the floor.

For example, what is to the right? Is that a wall or is it open to another space?

In your first layout, you have your cooktop only 18" from your sink. In both your island/peninsula is a barrier between the sink and the fridge.

When we cook, we take food out of the pantry/fridge, bring it to the sink to rinse, then prep between sink and cooktop. Additionally you have useless 12" drawers. And in your first layout, you would have to turn 180 degrees just to load the dishwasher. Think of everything dripping on the floor!

The right side of the plan is an exterior wall leading to the patio. The bottom door is to the dining room and the left door leads to the foyer/stairs up and down into the house.

My alternate plan with an open layout seems like a waste of open space. I agree with all of your comments on the island design, which is why I prefer the peninsula. I don’t think the walk to the fridge will be too cumbersome, this isn’t a very large kitchen.

Do you really consider a 12” depth cabinet a waste? I wouldn’t mind blocking off that section just to support the counter top.

Where exactly is the patio door? A fully-measured layout is easier to use. Similar to below with the widths of each wall/window/door/doorway and the distances between each wall/window/door/doorway labeled. Similar to below:

I would suggest you give Buehl a fully measured drawing and see what she can come up with.