Network Revived but File-Sharing Confused

The File-Sharing Network on my XP SP2 Home machines spontaneously
revived, but only after I did a lot of fumbling that seems to have
affected my ability to compare files of the same name that are shared
by two computers.

Here's my attempt to give a clear illustration of the problem:

Suppose I have two Work Group computers named "Desktop" and "Laptop,"
and also have two folders named "Alpha" and "Beta" on the C: drive of
each computer. I want to be able to compare and synch the two sets of
folders on either Desktop or Laptop. I can do that with "Alpha" but
not with "Beta."

In "My Network Places," I can see the icons for the Alpha and Beta
folders of both computers. There is a Key difference in the icon
names, however: Alpha is shown in Desktop as "Alpha (laptop)" and in
the Laptop as in "Alpha" (desktop.)" Beta, however, is labeled "Beta
(laptop)" on both computers.

What can I do to make the Beta folder function the same way as Alpha?
Would getting the name of the Beta folder on the Laptop to change from
"Beta(laptop)" to "Beta(desktop)" do the trick? Is that possible? If
so, how?

I get tongue twisted trying to explain this problem so I hope the above
is understandable.

I would appreciate any suggestions about changes I can try to make it
possible to view the two computers' Beta folders side-by-side on either
computer.

Thanks.