Ralph, you do understand me, you explained it dead on:
You have a parent site and a child site. The problem seems to be with the child site, right? You have "Home" button that is linking to the parent site rather than the child site, but you want the Home button on the child site to link to the child site Home page?
The only place you were off is that the Parent site is www.visproducts and the Child site is catalog.visproducts
If you look at all the button on the Child site, the URL they go to is the the "catalog", except the Home button which goes with that full URL back to the "www" of the Parent site.
In the Joomla admin, the Home button is marked as "default" and must be hard coded somewhere with an absolute URL. I just cannot figure out where it's rooted so I can change it. Googling around, the "baseURL" and/or "JURI" (in a helper.php file) I mentioned above is all I can find that seems to explain it, but those solutions don't seem to work for me.
I don't know if it's the same problem or not, but when I log into the Joomla Admin for catalog, every time I change and save something the URL ends up taking me over to the www Admin and I have to hit the back button and refresh to get it to properly take affect on the catalog Admin. Weird. That's why I think that somewhere in the system, at the very core of it, that a Default URL must be hard coded using the absolute URL.