You have the old Liftmasters, pre safety sensors. Unfortunately programming a universal can and will open the receiver, meaning anything within that frequency band can and will activate your opener, even if it's not programmed!
Sorry to say..time for an upgrade.
I've asked about this issue here before, perhaps you remember. I did try a couple of different universals, and they did cut the number of remotes I needed from six to three. Still too many for the cost of the universals. If I need three, I might as well have six. the good news is, when I took the universals back to the store and explained what was going on, they apologized and gave me my money back.
I think it a little "cheeky" that a manufacturer can cycle through generations of something like a remote control, making the new operators work with only new remotes, and leaving the older operators to work only with the old remotes, and when a customer complains, simply say "buy a new operator." My money comes to me a little harder than that, which is why my sun visor looks like the cockpit of a 747. I suppose there is a real reason why they won't make new operators backwards compatible with older remotes, I certainly hope they aren't doing it solely to sell more operators.
By the way, NONE of my operators are "pre safety sensors" they ALL have them. The operators may be old, but they aren't THAT old.
JBP
.