I'm not a programmer to understand the open source code... But do all brainwallet sites use the same algorithm to convert the passphrase to a private key and address? In other words, does a particular passphrase always generate the same private key on all sites?
I think the answer to that has to be
no. One convention is to use SHA256 (both bitaddress.org and brainwallet.org use this) but there is nothing stopping another site from using some other algorithm (for instance SHA256d, or including some salt).
As mentioned above, you're far better off using a deterministic wallet like electrum and saving the 12-word passphrase. Write it down (for your own sake, or for your beneficiaries in case the worst should happen, lost/forgotten passwords are far more risk than stolen ones) and keep it in a safe place or better, several.
And a general comment. Brainwallets are NOT safe for newbies to use, have a read of this thread
https://bitcointalk.org/index.php?topic=251037.0