RFCs & IANA specs about Ed25519 inaccurate?

While designing a crypto system based on existing standards and specifications i find myself questioning some of the accuracy in the current RFC-8037 and IANA specs around Edwards curves, 25519 and ed25519 notation

In RFC8037, they say that the public key in its compressed form (32 bytes) should be represented by X. But in Edwards curves, unlike Montgomery (Curve25519) shouldn't be Y with a signed bit for X instead?

Another thing that's odd is that on both RFC and IANA they put ed25519 in the elliptic curve section. But technically it's a signing scheme not a curve?

But in Edwards curves, unlike Montgomery (Curve25519) shouldn't be Y with a signed bit for X instead?

Yes

In RFC8037, they say that the public key in its compressed form (32 bytes) should be represented by X

They call it x in the JSON representation, which I agree is confusing. However, I've checked their example private key 9d61b19deffd5a60ba844af492ec2cc44449c5697b326919703bac031cae7f60. If you hash that with SHA-512, clamp it, scalar multiply it with the base point, extract the y co-ordinate of the resulting point, and apply the sign bit if necessary, the answer is identical to the hex public key example they list in the appendix.

I suspect their reasoning is that when x and y are specified, they really are co-ordinates of an EC point. However, when using 'point compression' to represent the point in a single 32-byte sequence, strictly speaking it's no longer a y co-ordinate because of the sign-bit manipulation. So I'm guessing they just decided to always place a compressed point representation in the x field and leave the y field empty (regardless of which of the two co-ordinates the compressed point representation was derived from).

Another thing that's odd is that on both RFC and IANA they put ed25519 in the elliptic curve section. But technically it's a signing scheme not a curve?

People commonly refer to both the signature scheme and to the curve itself as Ed25519. Monero, for example, uses the Ed25519 curve but not the Ed25519 signature scheme.