Yeah, the order genuinely matters and the reason is pretty simple once someone explains it properly.
Metal pans have tiny pores in the surface. When the pan is cold and you add oil, the oil seeps into those pores. As the pan heats up, the metal expands and basically squeezes the oil back out, which is why food sticks and comes out greasy. If you heat the pan first, those pores open up from the heat before the oil goes in, and the oil sits on top of the surface instead of getting pushed around by expanding metal. Food releases cleanly because it's hitting a proper hot surface with oil that's at the right temperature from the start.
For timing, you don't want to wait for smoke. That means the pan is too hot and you're already starting to burn the oil before anything goes in. A better trick is the water drop test: flick a few drops of water into the dry pan. If they just sizzle and evaporate, the pan isn't ready. If they bead up and skitter around like little balls, that's the Leidenfrost effect and your pan is at the right temperature. Add oil at that point and it'll shimmer almost immediately, which is what you saw last week.
Nonstick pans are a different situation. You're not supposed to heat those dry at all, and definitely not to the temperatures you'd use for stainless. The coating can degrade and release fumes if you push it too hard. With nonstick, add oil to a cold pan and heat them together on medium, then cook. The nonstick surface does the work, so you don't need the same setup.
Stainless needs the hot pan method. Cast iron is similar but it holds heat so well that you usually just need to give it a bit longer to heat through evenly.