Decide first what you NEED. Protection from sun or protection from sun and rain? (It's a "given" that whatever you build you would build to withstand the conditions of your region.) Are you committed to patio instead of deck? I notice your floor level is 2 or more feet above grade so keeping steps could become annoying. Are you considering off-the-shelf products only or site-built, too? Are your prices for the products but not installation (?)... that would make a difference. In general, a pavilion, which is almost a building, should be more expensive, complicated to build, and more likely to require a permit or higher level permit. A preliminary conversation with a local contractor who specializes in this type of work might be very enlightening for you. But having basic needs and general direction well thought out first would be good so the conversation doesn't get off track or.

Another thing to consider is that the off-the-shelf-looking items you're pondering look great when they're brand new and all their little details look special. But after they've sat out in the weather for a couple years or so, the little details start to matter less as wood turns gray and paint fades. If a pergola has a vine on it, one day the pergola disappears (except for the posts) and the vine becomes the predominant feature. If it were me, I'd rather have something a little more basic, but more stoutly built and longer lasting. Some of the details and lumber dimensions on that pergola seem petite.

Eric,
As a homeowner myself, I understand the need to be mindful of budget.
You probably already know this but the two different structures that you show in the photos above are not equal in value on several different levels. One is a well crafted structure and the other is a temporary DIY kit that will last for a couple of seasons at best.

I have to rebuild my own existing pergola, which was built from redwood about 40+ years ago. I'm going to rebuild with craftsmanship and value in mind and hope that it will last another 40 or 50 years. ( in my dream world I would love a solarium but that's just not going to happen .. too much $$ )

If you like the look of the wooden pergola in your first photo there are several ways to install a good looking water proof roof by using polycarbonate or plexiglass panels.

You may even be able to receive a energy saving rebate via your taxes or with your power company - locally ( CA ) I have worked on several projects where we built pergolas with solar panels and received energy rebates . They actually financially encourage people to install solar panels here to reduce energy dependence.

If you plan on staying in your house and want to add value , invest in the well crafted structure.
If you are looking for a temporary fix that will last for a couple of seasons then the kit pavillion might be the right choice for you .

How are the bugs? Around here, a patio or deck is practically useless for much of the time because it doesn't have screens.

Since you have neighbors looking into your backyard..
... there is an option to build a roofed pergola with its back against the fence on townhouse side. The walk to the kitchen will be longer, and the ground will have to be leveled (with a deck or a retaining wall) but this would give you instant privacy.

Thanks for the input, it does help. To address a few things- I'm considering a deck, my neighbor has the identical house (small 1950s ranch) and put a deck but his has a railing. I may not do a railing, instead wrap around stairs as its low enough local code won't require it. I plan to do a patio area in front of where the basement windows are and around the shed area. If I do this I'll have to do the pergola later.

As for sun- the house faces south west, so this side faces north east ish. The sun in the afternoon isn't that bad, it's mainly to create a visual break between the townhouse windows and my own house's windows and back yard. It doesn't have to be a compound with real high walls and complete isolation. Now that I think about it, a solid pavilion will block out light from the house windows, so that further leans me towards a pergola.

Bugs may be an issue as I had these dang biting insects in the past but I may have eradicated them by cutting down my overgrown bushes. My neighbors about 100 feet away have both a rear deck and a front porch and they regularly sit outside with no problem.

Also, I am an avid DIYer, so building something like a pergola is not out of the question. I just used the "off the shelf" examples but nothing is off the table now.

Another thing to consider is what you will actually USE. I think that, generally, people are more likely to use outdoor spaces attached to or in immediate proximity to the house than they are outdoor spaces they have to get to. Now, your yard isn't very big by the looks of things, but I'd bet you'd get more use out of the deck/patio/pergola investment than the gazebo investment. Gazebos are like English garden follies -- fun to look at, infrequently inhabited. So, put me in the deck/patio combo column.

An additional thought on the pergola idea. A pergola will work for your desired goals, but so would some plantings closer in to the deck/patio area. People tend to try to create privacy by planting walls of tall trees on the perimeter of property. In fact, you can get a good degree of (and often better) privacy with a mix of smaller trees and large shrubs planted closer to the area you're trying to obfuscate. A few specimen trees like pagoda dogwood or similar planted in the yard near the deck/patio would do the trick. Layers at different distances do as good/better a job of creating privacy as do green walls -- and have the extra benefit of being less confining in a relatively small, dark area to begin with.

Thanks for the reply. I did a real crude sketch of what I'm planning. Some of the dimensions may be off as I'm going by memory :) This is what I have planned. You're right the lot isn't that large, it's about 1/4 acre. Since most of the yard is grass, I wouldn't mind hardscaping/decking most of it to get more area.

Keep in mind, I'm a single guy in my late 20s, so this will not be used for kids etc. If anything when people come over or for gatherings, bbq's etc. The house isn't that big so expanded space would be great.

Obviously I haven't finalized the dimensions. I'll need a permit from the town too because the pergola involves footings etc. But first things first I need to do the hardscape and the deck.

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Blurry aerial view of my back yard:
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You've got a lot going on in that little drawing. Connect the patio to the deck (if you think you need both) -- ie you should step down from the deck directly to the patio. I would put the pergola over the deck/patio or not bother with it. As I suggested before, you could plant a few small specimen trees that would give you the layers needed for additional privacy along the edge of the planned patio/deck area -- there are plenty of choices that are readily available like japanese maple, dogwood, redbud, serviceberry etc.

I'll second everything that Mary has said - it has worked well for me in my own modest, overlooked backyard.
In the pictures below, you can see how the deck, pergola and patio work together...

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The best views in your garden are towards the shed. Unfortunately, the shed is doing nothing for this view. I would remove or relocate it.

You will have more privacy at ground level so i would build a small deck outside your door with quick access to a ground level patio towards the shed area ( with shed removed). Your pergola should be roughly where the shed is. Then you could develop a private garden area in front of the evergreens with long views toward the street.

My main point is to develop your plan where the best views are. And my vote is for the wooden pergola. There are shades you can get that are even rainproof. Fx shades are one of these.

Combine the deck with the pergola as Adrienne showed -- really like that!

I hate pergolas which are just plunked down in the middle of a yard without any context. I see that all the time in my neck of the woods and it just looks dumb.

agree about relocating the shed, too. Make those townhouse people look at the back of the shed.

Here is one that I added to my house in N. VA.