How deep does a waveguide need to be to be called a horn?
It's not a matter of depth, although in practice horns are usually deeper than waveguides.
"All waveguides are horns, but not all horns are waveguides" - Earl Geddes. Earl was, to the best of my knowledge, the first to use the term "waveguide" to describe a particular type of horn. And he designed imo superb waveguides.
My understanding is that a "waveguide" is a constant-directivity horn which does not use diffraction or vanes in order to achieve its constant directivity. In practice, a waveguide usually has much of its curvature way down in the throat, such that the curvature to transition from the compression driver's exit to the constant coverage angle of the walls of the waveguide happens very quickly. In contrast, most non-diffraction horns (exponential, tractrix, hyperbolic, spherical, L'Cleach, etc.) start out pretty much straight-sided at the throat and gradually widen towards the mouth, sort of like a trumpet.
At the risk of oversimplifying:
The main purpose of a waveguide (or we might call it a "waveguide-style horn") is to get constant-directivity behavior without any of the downsides of diffractive features or vanes. A waveguide's pattern is pretty much constant across the frequency spectrum (although in practice the pattern often narrows in the top octave as the driver itself starts beaming into a narrower angle than the angle of the walls). As the waveguide "funnels" all of the driver's output into a constant angle, the resulting frequency response is downward-sloping, so equalization is necessary. However once the on-axis response has been equalized, so has the off-axis response.
The main purpose of a non-diffraction horn is acoustic amplification by optimizing the acoustical impedance match between the driver and the air in the room, again kinda like a trumpet. Horns typically have a better low-end than waveguides, and typically have a higher efficiency, particularly on-axis. Horn crossovers tend to be simpler, because the on-axis response starts out approximately "flat". But the radiation pattern is narrow at the top end, gradually widening as we go down in frequency.
Diffraction horns combine some of the attributes of both, and often have good pattern control, putting the sound right where the designer wants it. But the trade-off is that diffractive features (slots and/or abrupt angles) introduce diffraction, and this diffraction tends to become increasingly audible and objectionable as the SPL increases. If you've ever heard prosound speakers that sounded harsh maybe you thought the drivers were being driven into distortion, but probably not - what you were probably hearing was the effects of diffraction.
I don't know much about the horns with vanes, but I think the shape of the vanes themselves are a way of combining a target expansion curve (exponential, hyperbolic, etc.) with relatively uniform coverage within the pattern width.
The reason why most waveguides are shallow is this: If a designer adopts "constant directivity" as a design goal, then they do not want a big directivity mis-match where the midwoofer or cone midrange crosses over to the waveguide. So they will design the waveguide such that its coverage pattern approximately matches that of the midwoofer or midrange in the crossover region, and then that same pattern is maintained from there on up. In practice you want to cross over your midwoofer or cone midrange well below its breakup peaks, so you end up crossing over where its pattern is still fairly wide (at least, wide compared to most horns). And in order to not have a directivity mis-match in that crossover region, the waveguide needs to have a similar pattern width. Waveguide pattern width corresponds with the angle of the walls, and a wide angle results in a shallow shape.
It is of course theoretically possible to make a truly narrow-pattern waveguide, but then you gotta deal with the radiation pattern width at the top-end of whatever is covering the region below that waveguide.
