Small correction from the source itself: Section 8 of the Bitcoin whitepaper says SPV is reliable "as long as honest nodes control the network" and more vulnerable if an attacker overpowers it. So it's strong evidence, not certainty.
What an SPV wallet does check on its own: the header chain and its proof of work, and a Merkle proof that your transaction is inside a block. What it can't check: whether the transactions in that block follow the rules. It trusts that the most-work chain is valid.
Privacy matters too. Old-style BIP37 bloom filters leak your addresses to the node you ask. Newer light clients use BIP157/158 compact block filters (Neutrino), where your wallet downloads a filter per block and matches it locally, so the server doesn't learn which addresses are yours.
The practical take still holds: a light wallet is fine for spending money, and a full node is for verifying everything yourself.
