Standard hash function of Linux
Though not my normal recommendation, the SHA-512-based password hash as implemented in Linux is a fine password hashing algorithm.  It is based on a well-understood cryptographically secure hash (SHA-512). It has a significant size salt and configurable number of rounds (default 5000). I would not feel insecure with such a hashing algorithm.

$2$ is indeed bcrypt, but note there are several variants around sometimes marked 2a or 2b and sometimes not marked. So, though bcrypt is considered very secure if you need a standard password hashing function, you need to be careful when choosing bcrypt.