therapod sitting has actually been studied.
It actually varies quite a bit depending on what therapod you are talking about, some theropods are nearly horizontal others nearly vertical. Raptors have a rear facing pelvispubuc boot (an enlarged section of the pubis that the body weight can sit on) meaning they actually end up with an upright sitting position. Early theropods are nearly horizontal while a t-rex on the other hand ends up sitting more in between. Birds actually vary wildly depending on the type of bird perching birds in particular do not sit.
forFor non-avian theropods in the sitting position they actually have 3-4 point contact, the pelvis (publicpubic boot) and both feet rest at the same level as the boot, with the tail on the ground or not depending on the species (most are a yes). You can look at the first image for the sitting position and the other two for the anatomy supporting that posture.
Because of this position a raptor chair is likely nothing more than a cushion on floor. Think along the lines of traditional Japanese furniture. By raptor I am assuming you means something like velociraptor or deinonychus and not an eagle. Note: however they need a lot of clearance behind them for the tail, in many species it is rigid for most of its length, in others especially the oviraptors it is shorter and more flexible. so unlike a bird they need need a lot of space behind them. For a vehicle there could be a lowered section of floor for feet forward and down from the seat, yielding something more like a motorcycle seat for the pubic boot to sit on.
A safety seat will likely have a flattened bar of metal running up along the stomach to anchor the torso against tilting forward, they can be strapped into that safely. Building true safety seats will be much harder than for humans. protecting the head in particular will be problematic since it is at the end of a long neck, whiplash will be a big problem. I can't think of a way to protect the head that does not prevent them from looking from side to side, the best I can think of is some kind of safety cable attached to the ceiling that prevents over extension. Really it depends on what you mean by safety seat, the seat for a car vs a rocket will be very different.
As a side note the sitting position is likely also the sleeping position, many birds tuck the head into the armpit to sleep and raptors definitely have the neck for it. the position and flexibility of the arms makes resting on the stomach unlikely. I can imagine tool using raptors inventing elevated wooden stands with a pillow on it for resting the head.
PS I know thistle's wrists are broken, but that is irrelevant for the discussion.