I just need to know is it correct to classify the
RSASSAPSS scheme and
sha[256 | 512 | 1]WithRSAEncryption all under RSA signature?
Yeah, of course, your RFC has references:
Although there are no known defects with the PKCS #1 v1.5 [P1v1.5] signature algorithm, RSASSA-PSS [P1v2.1] was developed in an effort to have more mathematically provable security. PKCS #1 v1.5 signatures were developed in an ad hoc manner; RSASSA-PSS was developed based on mathematical foundations.
If you follow that link you find that it was defined in RFC 3447, which is called the "Public-Key Cryptography Standards (PKCS) #1: RSA Cryptography Specifications Version 2.1". So these are the official standards from RSA Laboratories, except in RFC form. There is also v2.2 which just defines a different set of hash functions, of which only SHA-224 is moderately interesting in the given setting.
In full, the algorithm name is RSA (Rivest, Shamir, Adleman, after the team that thought it out), Signature Scheme with Appendix for SSA, and finally Probabilistic Signature Scheme for PSS. Commonly we just call it RSA-PSS or just PSS though.