It is an open standard by IETF.org
We can find the details in the mail archive of IETF, D. J. Bernstein's response;
It has become increasingly common for "Curve25519" to refer to an
elliptic curve, while the original paper defined "Curve25519" as an
X-coordinate DH system using that curve. "Ed25519" unambiguously refers
to an Edwards-coordinate signature system using that curve.
Kenny and others in Toronto recommended changing terminology to clearly
separate these three items. Let me suggest the following terminology:
- "X25519" is the recommended Montgomery-X-coordinate DH function.
- "Ed25519" is the recommended Edwards-coordinate signature system.
- "Curve25519" is the underlying elliptic curve.
All relevant coordinate systems already have standard names in the
literature, and I would suggest sticking to those names whenever it's
necessary to discuss the coordinate systems per se: [...]
Bolds are mine.