Someone advised me not to use any 'thin' wallet because they could easily expose my identity to 3rd party. I never heard about 'thin' wallets before.
What wallets are considered 'thin' and why are they insecure?
you should know that
security and
privacy are two completely different things.
thing wallets or SPV clients are
secure as long as it is a well known wallet like Electrum which is open source and many people are checking the code.
so i am sure that someone either made a mistake in using the term or you misunderstood him about security.
but on the other hand about the
privacy as it was said, they may not be the most private wallets since for example when you connect to an Electrum server to get your updated balances and latest blockchain headers your wallet will tell the server all your bitcoin addresses (only addresses never private keys or any sensitive data) so that server can log all your addresses and know these are yours and your IP is xx.xx.xx.xx