You are right that SPV as described in the Bitcoin paper does not explain how to discover payments to yourself, without downloading full blocks. My guess is that Satoshi either planned to develop this later, or that he assumed you'd just be told about payments to yourself (pay-to-IP, as existed next to pay-to-pubkeyhash as is used now).
Practical implementations of SPV nodes today however use a protocol extension called bloom filters, described in BIP 37. They use
getheaders to fetch block information prior to their wallet's birth timestamp, and request filtered blocks afterwards. To do so, they submit a bloom filter that describes the addresses and transactions they are interested in to the peer. The peer then only includes the relevant transactions in blocks it submits, together with their Merkle paths to prove that these transactions were indeed part of said block.