First of all, happy cake day.

Second, I'm pretty sure there isn't a third personality buried under the other two. They're either two separate personalities, or one combined personality at any given time. We probably saw Malachite as herself when she told dream Steven to "Go!" and for most of the Alexandrite fight. For instance: Defiantly smashing Watermelon Stevens, declaring "You think you can hold me down? Nobody can! Not anymore!" With the anger of Lapis and the bravado of Jasper.

If I had to guess, based on what little we've seen, the reason both components missed being Malachite is because, during their rare moments of cooperation, Malachite would've reveled in her own power. Her personality would probably be incredibly toxic. Complete disregard for others, an overwhelming desire to prove herself. No fear or anxiety whatsoever, except from someone else asserting themselves over her. Having felt that power, it's no wonder Jasper became so demoralized, so desperate to feel strong again. And having felt that unbridled confidence probably made Lapis feel a renewed sense of self-loathing.