The number of choices of clematis varieties is amazing and a bit overwhelming, isn't it!

Where are you? (minimum & maximum temperatures or zone and general location) What you can grow depends some on where you are.

You will want something for support for your clematis besides the two things you have mentioned. Clematis climb by twining their leaf stems (petioles) around narrow supports, so they won't climb anything that doesn't have something 1/4" or less in diameter for them to grab hold of such as wire or netting or string or a trellis. So you will need to add something with the fence and tree so they can climb.

How many feet/meters of fencing are you hoping to cover and how much managing are you willing and able to do? IME clematis by preference mostly grow up, but can be managed to grow sideways with a bit of guidance. The vine length varies considerably, so if you want to cover a lot of fence, choose varieties with longer vines, or plan to buy more of them.

You said, "would prefer large flower heads. Here again, just tidy up pruning." This would mostly put you in the type 2 (sometimes called type B) pruning group which mostly have large flowers and don't need major pruning, just a light prune after the spring flowering. Most of them will bloom again in the late summer or fall if you have a long enough season. Some of these are a bit fussier if you are in a hot, humid area like South Carolina or Florida from what other experienced growers have said on this forum, though that isn't an issue for me. In my cold winters, I find that usually the type 2 vines get killed to the snow line, so I only get one bloom in late spring on the new growth. My season is too short for a second bloom.

If you are in a warmer area than I am, some of the C. montana selections might work for you. They are quite large and don't need any pruning (type 1.) Flowers aren't huge, but aren't tiny either, and the vines can get long. They bloom in shades of pink and white.

The type 3 prunes are generally the easiest to grow since most of them are quite vigorous. Pruning is easy (hard prune down to 2 or 3 sets of buds per stem in late winter/early spring) and they regrow quickly to their full height in spring. In my zone I find that these are my most floriferous, usually blooming for several weeks during the summer.

So, to help you limit your selections, think about what type of pruning you want to do (none are difficult IMO), how long you want the vines to be, and which you are able to grow. Think about when you want flowers. For color, think about what would look nice against the terracotta fence to your eye. I'd probably go with whites and blues, but others might like pinks and reds, so it's really up to your preferences as far as color.

One you have answered some of these questions and know color preference and pruning type, look around in your area (local and online nurseries) to see what is available for purchase. IMO that is easier than looking on line at lists of plants and then not being able to find one that you've fallen in love with photos of. Once you know what is available for you to buy, then go to the website Clematis on the Web. If you type in one word of the name (the search form doesn't like spaces) and leave everything else blank, you will get a link to a page just about that clematis that includes pruning type, photos, flower size, and vine length. Then just choose a few that fit your needs. As far as I am concerned, all clematis are gorgeous and you can't really go wrong except by delaying planting for more years. If you are really concerned, come back with a list of plants to see how others have fared with them. The reason not to start with C on the Web is that many of the plants there aren't readily available in commerce or may not be available on your continent, so it will be frustrating to find many of them if you start from their lists, though you can do that.

Come back and let us know what you've decided or if you have more questions.