As far as I know, though, there isn't any means of transmitting malware through the Tor Browser that doesn't work on standard Firefox.
Note that, in this case, users' computers weren't infected. The exploit only leaked their real IP addresses, which allowed them to be identified and located. Many "Tor specific" attacks are of this nature, as exposing a user's real IP address is not an exploit at all if that's the IP address they're using publicly.
In regards to the page telling you their "exploit was successful", it's much easier to type:
<p>exploit was successful</p>
Than it is to exploit a hardened up-to-date Firefox browser. It's possible you got infected, but you're probably fine.
And, to answer your last question, yes, Tails is safer. It attempts to route all Internet traffic through Tor, so even some exploits that get by Firefox will still not access the real IP. However, as with all security software, Tails does occasionally have exploits of its own.