There were sites around back in 2010 that allowed you to by bitcoins eg, New Liberty Standard, Bitcoin Market, MtGox
Some discussed here: https://bitcointalk.org/index.php?topic=1800.0

You would have needed a bitcoin address for them to transfer the bitcoins to.

Usually you would download the bitcoin client software and use that to generate an address that the coins could be transferred to.

Otherwise there were online address generators such as bitaddress that allowed you to make wallet addresses and print out paper wallets. They mention a PGP public key.

Here's the earliest version i can find in archive.org that works.
https://web.archive.org/web/20130420072538/https://www.bitaddress.org/bitaddress.org-v2.4-SHA1-1d5951f6a04dd5a287ac925da4e626870ee58d60.html

That way you could have a bitcoin address without ever having downloaded the bitcoin client.

They have a brain wallet generator, but that wasn't added until 2012 so unless the herald didn't report on the pizza buying incident until later maybe your dates are off? Memory can sometime be inaccurate. Earliest article I can find on the heralds website is from 2013, but of course that doesn't mean they didn't report about it earlier, just that it's no longer findable on their website. https://www.heraldsun.com.au/news/national/bitcoin-fervour-goes-viral/news-story/ea5cfc758513db09169163ea94067cab

Otherwise there were online wallet services like blockchain.info (now blockchain.com), MyBitcoin https://en.bitcoin.it/wiki/MyBitcoin , Instawallet https://en.bitcoin.it/wiki/Instawallet etc

Do you have access to the email accounts you were using back then?
Searching them might find some clues as to what sites you might have used.