Which also makes me wonder how many of the shooting stars I've seen recently are just old starlinks burning up.
You probably just thought it was a star or a plane. They move but relatively slowly (even a fast LEO sat will cover the sky in about 5 minutes). They look just like a star apart from moving slowly. Depending on angles they can look pretty dim, especially the latest SpaceX sats. But the ISS is usually really bright because it's so huge and technically it's also a satellite.
You can tell them apart from a plane because they don't flash.
Edit: But yes there are several conditions that need to be met to see them like the other posters have mentioned. But every clear night near dusk or dawn you will see sats for sure. There are just so damn many in LEO now.
The flashing of satellites will generally be gentler than the short burst strobing of an aircraft.