The concept of "the Other" as a replacement for the Beast appears in the Vampire: The Requiem Chronicler's Guide, published in 2006. Details for this variant appear on pages 59-65, but it's summed up in the introduction like this:
The Other wonders how Kindred would find eternity if the Beast weren't a ferocious, ravenous devourer but, instead, an insidious and devious schemer.
In brief, the Other presents as a second personality, a voice in the character's mind, always criticising the vampire's humane actions and trying to persuade them to kill and feed. It works through dialogue, not emotions or instinct like the Beast does, and allows for more finesse and precision in presenting the temptations of the vampiric condition. The Other's priorities are satiation and personal safety, and it tempts characters with promises of secrets about vampire history and abilities.
The rules differences include removing the Predator's Taint, replacing Frenzy with Imprisonment (the Other takes control of the vampire's body, with various advantages to the Other), and changing the terms and effects of Riding the Wave (which becomes "Allying with the Other") and reaching zero Humanity (effectively permanent Imprisonment).