It's physics of lithium batteries, yeah, but here's the thing: capacity loss in the cold isn't permanent damage, it's just a temporary drop in performance. Once your phone warms up indoors, the battery will "recover" most of its charge. That's why an iPhone shows 20% in freezing weather and shuts down, but then you turn it back on inside - and suddenly it's at 40%.
But there's an important threshold. Subzero temperatures freeze the electrolyte inside the battery, viscosity increases, ions move slower, and internal resistance skyrockets. In the cold, it physically can't deliver the current it needs to - that's why it dies in an hour. If you regularly put the battery through deep discharge cycles in freezing temps (when your phone shuts down because it "thinks" it's dead), that'll cause real damage over time. One or two walks in the cold - no big deal, but if it's every day - that's different, the battery degrades faster.
Practical tip: if you spend a lot of time in cold weather, keep your phone under your jacket close to your body or in an insulated case. And don't let the battery drain all the way to zero in the cold - better to head home with 30% than kill it outside.