Core Shaper isn't actually all that interesting/good/hated. Magnum Opus probably being the only one that gets singled out, but Diesel and The Maker's Eye are other cards from Core Set that really set the tone of Shaper. Shaper is about Drawing Cards, runs on R&D, and Programs to solve problems. (Edit: Notably, Shaper is the only way to get extra MU outside of a Console in Core Set. Unless you count Anarch's Djinn, anyway.)
Anarch has Parasite and Datasucker, as well as Yog. Corroder got quasi-hate for just being the Best Fracter for the longest time. It was more held up as a Gold Standard of icebreakers, but contributed to Barriers being the weakest of all ICE types. Anarch honestly had a lot of the 'benchmark' cards. 'Can your code gate stand up to Yog?' 'How easy is this Barrier to break with Corroder?' 'You can break this Sentry with Mimic.'
Criminal in Core set has Account Siphon and Inside Job, and that's about it. Bank Job and Desperado also help to define the faction: We're about high-impact single runs to make money, and we also get paid to run successfully.
As stated, HB and Jinteki didn't get hugely impactful cards in Core. The defining cards for HB are Accelerated Beta Test, Archived Memories, Biotic Labor, and Adonis Campaign. HB is about ICE, Heap Recursion, Extra Clicks, and Econ Assets. Jinteki has Snare!, Personal Evolution, and, uh... Data Mine? (Neural Katana for a little while.) Look, Core set Jinteki was really very weak and relied on Runner messing up. You could say their faction identity at that point was about letting Runners through their servers; for a price, and capitalizing on Runner misplays.
Weyland's one card from Core that gets hate is Scorched Earth, and it's more because of what it causes Tags to mean. Tags, by just the game rules, means that the Corp can interact with Runner Resources directly. Sure, this hurts Runners, but in theory if you have no Resources, you don't care if you're tagged. (See: Criminal Tag-Me with Account Siphon.) Scorched Earth changed that game. Now, you can actually just straight lose the game if you don't clear your tags. And even worse, if there's any way the Corp has of tagging you on their turn, you can straight lose the game. Other cards for Weyland that defined the faction are Hostile Takeover, Archer, Ice Wall, and Beanstalk Royalties. The faction is about Making Money, at any cost, ICE that gets progressively more difficult to break if you just sit back and let Weyland do its thing, and Forfeiting Agendas. (They had the first Forfeit mechanic in Corp. Data Dealer existed in Criminal for the Runner side.) Oh, and blowing up your apartment block. They do that, too. One thing about Weyland is stated by their conspicuous lack of a card: In Core, they're the only faction without an Ambush asset. If Weyland does bad things to the runner, they're public about it. They don't really do Sneaky. (Aggressive Negotiations also set the stage for Weyland being the 'tutor' faction, though the specific card was weak.)
NBN's Core cards are numerous. SEA Source and Breaking News together say the faction is about Tagging you, on their terms. Data Raven and Tollbooth say that the faction is about doing bad things to you, regardless of whether you actually broke the ICE or not. (Matrix Analyzer also carries that theme, but its effect is poor.) Anonymous Tip says the faction is about Drawing Cards, similar to Diesel from Shaper. Closed Accounts and Psychographics combine with the earlier cards to have something to do with those tags that they're giving you. Because as stated earlier, the only default thing is trashing Resources, and that doesn't work against all runners. SanSan City Grid and Astroscript Pilot Program combined to say this was the faction that cared about actually scoring agendas quickly. This is contrasted against HB's Biotic Labor. HB just wants extra clicks, NBN is specifically about scoring agendas faster.