I am not a fan of a zillion can lights in the ceiling.

Why not get a good pendant light for the living room -dining room area, and then you can get a floor lamp or torchere, something like that for the corner where the sofa is, or behind the chair, where the person sits who does the reading in the house.

If I am reading your drawings correctly, that room will get plenty of light through the windows during the day.

I would also get a ceiling fixture (either hanging or not) for the kitchen and the adjacent entry room at the back door, these might be matching.

The one in the living room I would suggest be a pendant fixture not matching the one in the kitchen but in the same family, and larger.

Since you island is halfway into your dining room/living room, I don't think it would look appropriate to have hanging lights over it, and the island will be adequately lit by the two fixtures you will already have in that space.

If you want to put lights underneath the kitchen cabinets, go ahead.

I've never had them and never felt that I needed them.

Have you felt that you can't see what you are doing as you prepare food in the kitchen, or are you doing this because you think it is comme il faut these days?

I am wondering about how close the island is to the door.

How wide is the aisle between the island and the sink? Can you move the island closer? Or rotate it 90 degrees?

IMO pot lights are the best for overall kitchen lighting thet need to be placed to light the work area without shadows , under cabinet lighting is also nice Pendants over the island need to light the island top not the ceiling. All bulbs IMO should be LEDs in 4000K and those pot lights no bigger than 4” cans An actul chandeier is what you need over the dining table and I personally do not like pot lights in DRS or LRs In your LR 2 table lamps and a good floor lamp will light that samll space.

Doesn't pendant light mean chandelier?

Anyway, when I said pendant light over the dining room table, I meant a chandelier.

By pendant, I meant something that hangs.

And how big is that space, by the way.

There are no dimensions.

Patricia is correct that table lamps are always great in a living room, that is what I have, but it looks like (if your drawing is to scale) that your side tables are too small for table lamps, that is why I suggested a torchere. Oh I do love a great vintage one. Though that may not be your style at all. Well, they have really nice floor lamps, and a chandelier over the dining table.