The BSD socket API on many systems implements a raw socket type[1], so you can use the socket APIs to talk raw IP.

Some systems (Linux, Darwin) also implement a special icmp socket type which can allow unprivileged ping.

[1]: https://man7.org/linux/man-pages/man7/ip.7.html

Regardless of what layer ICMP works at, its traffic is still transmitted using datagrams, and the call to do so is socket(), yes?

Are you just trying to point out the misnomer?