Mindfulness meditation is very explicitly not about turning thoughts off, but about letting thoughts arise and fall away without getting caught up in it.
Concentration meditation practices do involve turning thoughts off.
That said, I also find the comparison to CBT curious.
> (in 95% of the cases it's your body).
In mindfulness meditation, while it can be your body, it is not generally nearly as dominant a focus unless you want it to be. The "baseline" tends to be mindfulness of breath (anapanasati) because it is a convenient basis for a necessary level of concentration, and while it may involve the body peripherally as where you will notice the breath, is not about the body (indeed, while mindfulness of breath is simple as a starting point, people have written 500+ pages of books just on that one subject, and that involves e.g. the selection of other meditation objects to use while practising mindfulness of breath to achieve specific effects, such as dulling distractions, and a lot of that seems to be down to draw attention away from your body, because your body is a rife source of distracting sensations).
Mindfulness of the body is a separate aspect which tends to explicitly involve letting the sensations of part of the body fill your mind and diffuse outwards until it feels like that part of your body is all there is. When you do it with sufficient concentration it produces very curious sensations. It can also be a useful practice to reduce physical pain (I've had success with it but to very varying degree) - I see that as a trick of the mind where you basically seek to "spread the pain around" mentally so that it feels like it dissolves; sharp pains feel like dull aches. Nevertheless when successful it is a very useful trick of the mind (as a skeptic it feels to me as intentionally seeking out a placebo effect, but if it works it works).
Mindfulness of mind and and mindfulness of emotion are other separate aspects, and purely disassociated from the body, though you will tend to use mindfulness of breath to bring the concentration needed for it.
Most beginners mindfulness practice tends to focus first on mindfulness of breath to attain base levels of concentration and then mindfulness of mind.
If you focus only on the body as the object of meditation, you miss out on many aspects.