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Looking for a landscape lens, considered the following to couple with my 5DIII:
Canon 16-35 L II
Tokina 16-28 f2.8
Somehow I feel the TSE would be a bit cumbersome for me, but that's just a feeling. So I would go for zoom just for convenience.
You have presented a big question inside a smaller question. The smaller question has to do with choosing among these lenses as tools for landscape. The larger question has to do with whether or not wide to ultra wide lenses (as these are, at last on your full frame camera) are necessarily "landscape lenses."
I'll take the bigger question first, but keep the answer short. The answer to the question of what sort of focal lengths are "landscape" focal lengths is very subjective. There is a type of landscape photograph that is very familiar to us - what I've heard referred to as the near-far image. It usually features some foreground point of focus - a rock, a flower or group of flowers, a plant, etc - with more distant features - mountain, sky, clouds, etc - beyond that. The familiarity of this type of image, and it can be an effective type, has led some to assume that this is how landscape is done.
In my experience, it isn't that simple. Many landscape photographers I know more often shoot this subject with longer focal lengths. In fact, I can name several who will tell you that their most-used lens for landscape is a 70-200mm zoom. I know it is for me. There are four zooms among the lenses that I carry for landscape photography... and the one I own from your list is the one I use the least for this sort of subject.
I mention this on the off-chance that you are looking at these three lenses, all in the ultra-wide category, because someone told you that ultra wide is what you need for shooting landscape. If so, you might want to consider a different or an expanded list. On the other hand, if your extensive experience shooting landscape has taught you that your work is most often done with ultra-wides... you can ignore this and the previous paragraph. ;-)
Among those lenses, if your primary or nearly exclusive goal is to shoot landscape, I would almost always go for the 17-40mm f/4. Let's assume that your landscape photography is more of the common variety - shot at small apertures and working in decent light and from the tripod. The 16-35mm lenses are fine lenses - both of them - but their chief virtue is their better performance wide open - they get to f/2.8 and they are better at f/4 - when shooting handheld in low light conditions. That would be quite atypical for most landscape shooters.
On the other hand, the 17--40 is as good as the 16-35 lenses (perhaps marginally though inconsequentially better in the center) when stopped down to typical landscape apertures and working from the tripod. It is also lighter, smaller, and less expensive, and it takes the more typical 77mm diameter filters that you might need for other L zooms. (There are exceptions here - a few more now take the larger and more expensive 82mm diameter filter that the 16-35 II uses.)
If you are convinced that the 16-35mm lenses are what you want for landscape - and they'll do fine for it, though they won't offer any advantages for landscape - the case for the more expensive model II is weak for your use. The main reported virtue of the newer version of the 16-35mm lens is very slightly better performance at (but not beyond) f/2.8. Unless you are an unusual landscape shooter, that hardly seems important. And the older version uses the smaller 77mm filters.
There are a few situations in which a TS could provide some advantages, but few landscape photographers that I know will substitute a TS for an ultrawide zoom. The TS allows you to tilt the plane of focus, most often to bring near and far objects into focus at slightly larger apertures, thus relying less on the larger DOF of small apertures. But this is less of an issue with an ultrawide lens than on longer focal lengths where the DOF is naturally smaller. Another thing you can do with the TS is correct for converging lines, for example trees that tilt inward when the horizon is low in the frame.
But there are downsides, too - or else lots more people would use them. Most people will regard them, as you seem to, as cumbersome by comparison to using zooms. They are larger and heavier and more difficult to focus. They are, despite their fine optical performance and ability to tilt the focus plane and deal with converging perspective lines, less flexible in many other ways.