You must show it. Either from a distance (but not too much) or take 3 slightly overlapping shots that pan the scene from left to right.

The zone is only part of the story. What region? I think that guides the style of planting as well. The larger context.

Ah... so to me, that evokes rhododendrons,azaleas, camellias,magnolias...

What is that fence situation? You and your neighbor each put up one a few inches apart?

Now that there is a competing dominant fence, it nullifies yours. In spite of the fact that you have the right to it, I would remove it because it makes things worse for you. While removing it wouldn't change the need for different planting, it would nevertheless help with maintenance and appearance.

I presume that where you need the privacy/screening most is in line with the patio...? A large shrub form (with foliage to the ground) would take care of that. A big, fat Camellia, for example, would work, and be maintainable over the long term. Of course, you could explore using other evergreen shrubs. Extend the shrubs as far as the screening is needed

I would section off the entire space between patio and fence (as shown in illustration) and fill it with a solid bed of groundcover so that there's no mowing this area at all. Something like solid 'Big blue' Liriope would be nice since it blooms for a period during the summer and looks tidy otherwise. But there are many things you could use.

Also, there could be an odd-number grouping of small, multi-trunk tree form plants such as crape myrtle to give a filtered, semi-screened view along the property line. The more trunks there are, the more screening effect.

And finally, some annuals or perennials for color near the corner of the house could give some cheer to the patio area.

You shouldn't mistake what I'm offering you for a plan. A plan is essentially a map that shows how things are laid out and where everything goes. From a scaled plan you could calculate quantities. Here, you can't. This is just a concept -- an idea that you would convert to a plan and improve upon it along the way, working out all the details and tidying up wherever I've left things messy.

I had zero shade in my backyard and needed some fast, so planted four right off the bat. You will be surprised how fast trees will grow back there if you pick the right species. Three to five years and you'll be cutting them back! I can't think of any tree that you could plant that will offer immediate shade, that's what patio umbrellas are for. :)

Camellias can take a lot of sun ... more than they are given credit for. I found this snippet on an old GardenWeb post: "I have seen very large old (100 years!) Camellia japonicas in Pasadena (very hot summers) that have gotten tall enough to reach full all day sun; they look great. Their roots are completely shaded, which likely makes that possible."