Well "wallet" as storage of keys should always be under your control. There are online wallets for higher convenience, but as the old saying goes "not your keys, not your bitcoin". It's probably a bad term, but I'd rather say "local (SPV) node" instead of "wallet".
When you are sending there is no problem with malicious peers as the transaction is (usually) cryptographically signed, so the most they can do is not relay your data to the rest of the network (DoS). Like a regular bitcoin node you can
basically just send the tx out to multiple peers.
The important part where blockchain comes into play is verifying whether a transaction actually happened (possibly also the one that you've submitted).
In this context "centralized" means you basically just ask another node. Now the node will usually tell you the truth unless it has some incentive to cheat. Like owners want to persuade you they've sent you money in exchange for your goods or services when in fact they didn't. You could ask multiple nodes, but you can't be sure they're not colluding to scam you (same goes for block explorers).
In order to be 100 %, you'd need to download the whole bitcoin blockchain from genesis and check for yourself. Point is that you do it from multiple nodes and due to PoW they can't make up that data from thin air. Downloading the whole chain is however a bit impractical so what "simplified payment verification" does is basically allows you to just download the chain of block headers (that are only 80 bytes each). Then you see that the block containing the transaction is part of the longest chain (and that means that there was consensus on the network that the transaction you are wondering about was indeed valid).
