Over the years I've had a few different kinds of projectors handed down to me.
I had some old Yankee carousel tray clones. These were probably what your parents had if they had a Kodak Carousel projector. The oldest 500 series projectors could only take 80-slide trays. Projectors that can use 140 trays can also use 80 trays.
Another popular tray system is the Sawyers. The circular tray sits vertically (more like a ferris-wheel), has zigzag teeth and the slides are held in the slots with springs. The Sawyers projector could also take rectangular trays.
Revere made another rectangular tray system that was incompatible with Sawyers.
ps I work for Kodak but the opinions and positions I take are my own and not necessarily those of EKC.