Some authors draw a distinction between:
- RSA-PSS, analyzed by Bellare and Rogaway in 1996 and proposed for IEEE P1363 in 1998, which is roughly defined in terms of $H(r \mathbin\| m)$, and
- RSASSA-PSS, standardized in IEEE P1363-2000 and RSA PKCS #1 v2.1, which is roughly defined in terms of $H(r \mathbin\| H(m))$.
This distinction is significant: RSA-PSS relies only on target collision resistance (or universal one-way hash function…ness) of $H$, while RSASSA-PSS relies on full collision resistance of $H$.
Another way to view it is that RSASSA-PSS—like RSASSA-PKCS1-v1_5 and any other signature scheme where the message figures in only via $H(m)$—is vulnerable to collisions in $H$. This spells bad news when $H$ is MD5, as many people used in practice for many years. Academic cryptographers publicly demonstrated certificate forgery using an MD5 collision attack on RSASSA-PKCS1-v1_5; the United States and Israel forged a Microsoft software update signing certificate using an independently developed MD5 collision attack to sabotage Iran's nuclear program. But as far as anyone can tell, MD5's target collision resistance still holds up to this day, so RSA-PSS would have thwarted this entire avenue for certificate forgery even with MD5.
Why IEEE P1363-2000 and RSA PKCS #1 v2.1 standardized $H(r \mathbin\| H(m))$ instead of $H(r \mathbin\| m)$ is a mystery to me. Coincidentally, this standardization happened around the same time that RSA, Inc., accepted a 10e6 USD bribe from the NSA to deploy Dual_EC_DRBG to all their RSA-BSAFE customers.
Modern signature schemes like Ed25519 use the design principle $H(r \mathbin\| m)$ in order to rely only on the weaker property target collision resistance like PSS or at least enhanced target collision resistance, and as such are advertised to have collision resilience. (More on the history of these and related concepts in a previous answer.)