They are very fluid terms, and classic does refer to paranoia back in the day. However the way paranoia works is this:
Usually the first time people play paranoia [unless the dm/gm pushes for it] they play zap. After awhile they slowly end up becoming more and more 'straight' in gamestyle if they play lots of games.
It's more of a 'spectrum' then hard and fast rules.
Zap<->Classic<->Straight
Where the original design for the game is 'classic' but in-reality new gms/dms/players often devolved into zap before the creators of the game even called zap a playstyle... so even if people often try and play classic they end up playing zap. Likewise because i've played/gm'd quite a bit in the last 6 months, my style of play/style of dming even if i call it classic is likely sliding into straight.
Edit: i should also mention i'm using the paranoia XP manual, which is i think, the last 'all-around' instructions manual. It's entirely possible the early ones had different/contradicting opinions.
Edit2: after reading more on it in the manual, i strongly lean towards straight.
From the manual:
PARANOIA Straight is a darkly satiric style emphasizing tension, mutual suspicion, spying and subterfuge, and careful collection of evidence. Alpha Complex is an oppressive totalitarian bureaucracy that works scarily well, using its own insane but comprehensible logic. The all-powerful Computer, though willing to listen to reason, is always four steps ahead of the players. Troubleshooters are generally esteemed as heroic adventurers. Troubleshooting is a high-risk, high-payoff path to social advancement, and those who play it smart and low-key can learn to survive in the machinery of oppression. Troubleshooters aren’t automatically rivals from the first minute, but gradually develop personal enmities and evidence dossiers. This style can be extremely interesting. Using the designated Straight rules with experienced PARANOIA players, you can keep characters alive long enough to play a ‘miniseries’, a linked set of missions with continuing characters. Avoid building ready-made rivalries into the PCs’ missions; they’ll develop these themselves. Players immerse themselves in the setting, learn the ropes and develop emotional commitment to their characters. In such a setting you can scare them thoroughly, toes to eyebrows.