If I read this right, you have saved files to the floppy drive on your computer and, if you put the disk back into that same computer, it can see them all and read them correctly. The problem is isolated to another "public" computer which cannot read the files.
By far the most common problem with floppy systems is that the heads accumulate dirt and can't read and write properly. The solution, as sminlal suggests, is to get a floppy disk head cleaner kit and clean them. Just remember that, since the process uses liquid alcohol solvent, you should let the drive unit dry out for a few minutes before trying to use it.
I've used floppies for years, but often on machines where they are used only infrequently. By experience I have found that there is a hazard in this process. If you put a perfectly good floppy into a dirty drive it is possible for the dirt on the heads to scratch the disk's surface and permanently damage the medium, making the file there inaccessible. If that happens to be in the root directory or FAT areas the whole disk becomes useless. For that reason I'm in the habit of carrying a floppy drive head cleaner kit with me when I use them. I start by assuming that the drive is dirty and clean it as the very FIRST step, then let it dry out. Oh, even before that step, I will bend down, push open the floppy drive's front cover, and blow into the slot to try to remove any really loose dust before starting with the head cleaner. Only when the drive has been cleaned and dried out do I insert my floppy disk and use it.
If the disk you are using is still fully readable in the drive where it was written, then at least it has not been physically damaged. Try cleaning the "public" machine's drive unit, then see if it works.
There is still possibility this will not work for you. It is possible the "public" machine's drive is malfunctioning or its connections are dirty. And rarely you run into a machine in which the heads are so misaligned that it works with floppies written on it, but not with floppies written on another machine.
A "public" machine can have lots of problems with floppy drives and dirt. Usually they are in places where there's lots of dirt and dust in the vicinity, and they are NOT routinely maintained. Especially for this old technology, it is likely nobody has bothered to look into the drive, clean out its gross dust accumulation, and clean the heads. There is a design characteristic that makes this more likely, too. Many machines have their fans set up so that there is actually a net small vacuum inside the case and the exhaust fans are sucking air into the case through every opening. Well, one such place is the slot (even the slot that appears to be covered with a spring-loaded flap) in the floppy drives' front. So lots of dust-laden air is drawn in here and must make its way through small spaces and around components, leaving dust deposits there.