I agree with the first point, but I think compressing and non-reversible are necessary conditions for a given function to be called a hash function; if they weren't, any mathematical function would do, wouldn't it?
One could see a hash function as an (extremely) lossy compression method. However, lossy compression only makes sense when you can exploit features of the domain, e.g., psychoacccoustics with sound, or characteristics of human vision with photos; perceptual hashes come to mind here.
It's really quite simple. Perfect hashes exists and are hashes. Perfect hashes do not, as a matter of definition, compress. They typically aren't reversible, because it's not an useful feature for a hash, but they could be - and if nothing else, they can always be deterministically brute forced (as they have no collisions), which is a (very bad) form of reversibility.
It's not meaningful to study hashes as compression, since compression is by definition reversible.
One could see a hash function as an (extremely) lossy compression method. However, lossy compression only makes sense when you can exploit features of the domain, e.g., psychoacccoustics with sound, or characteristics of human vision with photos; perceptual hashes come to mind here.