Your run looks really nice and well-constructed, and having the coop inside the run (especially since your run is covered) can work very well. I would just build a new coop similar to the old one, or even just make some modifications to your existing coop (if it's in decent shape and parasite-free,) because it looks like it just needs a few easy changes:
1. More ventilation - your chickens might prefer to roost outside because it looks like the coop has no ventilation except the big door opening. You could cut the siding about 6" below the roof on the low side, make it the same height all around, and use hardware cloth from that height up to the roof on all sides.
2. Put the coop on much longer legs, like 3'-4' feet off the ground. This way, the bottom is at the height of your waist, making it much easier to clean. You can fit the front of a wheelbarrow under it, and just pull the dirty bedding out directly into the wheelbarrow.
Also, it makes room for the chickens to go under the coop, and you could keep their food, water and a dust-bath under there. If the coop roof is about or close to the height of your run, it will encourage the chickens to roost in the coop instead of on top of it.
3. Roosts: We can't see from the pictures where the coop has a roost inside, but in the "similar but new" coop, make the roost above the height of the nest boxes but at least a foot below the ventilation, and make a ramp for the chickens to walk up into the coop. They feel safest roosting as high as they can get off the ground, so walking up a ramp to go inside, then jumping up to a roost, will encourage them to roost inside.
Take down the outside roosts and metal framework, so they have no excuse to avoid roosting inside. If they need more entertainment outside, put in a few logs or stumps to make different "areas" in the run.
Depending on the climate you live in, you can likely take down all the plastic from the run, or maybe just cover one or two sides where the wind comes from if your area is very windy or consistently gets below freezing.