What makes an overdrive feel alive rather than just loud?

Some overdrives seem to respond to how hard you dig in or how you roll back your guitar volume, and others just sit at a fixed color no matter what you do. Curious whether people think that responsiveness comes mostly from the clipping topology, the input impedance, or something about the gain staging that's harder to pin down. The pedals that actually change behavior based on your playing feel like a different category entirely.
Source: Powerful Tone, Tiny Price Tag: Best Cheap Overdrive Pedals - Gearank - Music Gear Information for Live Performance or Studio ↗