I have installed MC32. I tried to explain in detail back on June 18th what I was having trouble with. I'm not sure the easiest way to have the same "build" on the 2 laptops. I have no idea how or why they are different when I did them together. One was a new install and one an update from MC25. I find this very difficult and getting a direct answer is the worst part. I have used 3 of my restores on this and it feels like I am not getting anywhere. Folks that are not very proficient with computer usage sometimes need a little more direction. I am very confused at the point. I do not see what another restore will change.
If you are still having minor version differences (32.0.47 vs 32.0.55) between your two pcs, it seems likely to be that you have them set to different update channels in Media Center. You can check this in Media Center by clicking Help - Update Channel then change them both to latest for now then click on the Help - Check for Updates option to ensure that they both move to 32.0.58 (current as of this message). If you want to use only Stable minor versions after this update, you can change both pcs back to the Help - Update Channel - Stable. Please note that minor version differences between the server (the pc whose library you want to use) and the client (the pc who will "pull" the library from the server) don't typically cause issues in recent years.
If you're no longer having issues with your "old library" on the "old laptop" that has been updated to Media Center 32 (beyond any build differences referenced above), it should be your "server" pc that shares it's library with the new pc (so that the library is the same between both pcs). On your old laptop it seems like you have already enabled the Media Network/DLNA option and have the access key that will need to be entered into the "new laptop" in Media Center. In order for the new laptop to be able to use the library from your old laptop, Media Center must be set in Tools - Options - Settings - Windows Startup to be Media Server on both pcs. Note that Media Server can only run a single instance per pc so if you have more than one user profile, only one can be in use at a time with Media Center/Server or you'll get errors.
Once the old laptop is configured as mentioned, you would go to the new laptop then ensure that you have enabled the Media Network/DLNA options and have told Windows Firewall when it pops up to ask that yes Media Center needs to to connect to public/private networks (this had to be done on the old laptop in order to get the access key from the JRiver servers). Once this has been done, you should be able to click File - Library then see a list of libraries available then select the one that starts with your old laptop's name on the network followed by (Library Server). Once you click on the old laptop's library, Media Center should tell you loading library. Once it has loaded it, you should be able to use the library from your old laptop to play on your new laptop independently. If you don't see the old laptop library, I suggest rebooting both laptop to ensure that only one user profile is running on each and that Media Server is running on both; once this has been done, please open Media Center on the new laptop then check for the library server as mentioned; if it still doesn't show for some reason, you can manually select it by clicking File - Library - Library Manager then clicking Add Library. On the New Library screen which will appear enter the name that you want to use for your old laptop for clarity then select Library Server then enter the access key from your old laptop and click OK. This should then load the old laptop library for your use on the new laptop.
In order to keep things as simple as possible and not need to add username password / https options, you will not be able to make any changes to the library contents/playlists/etc. on the new laptop but can on the old laptop. Changes to the library on the old laptop will require the new laptop to have both Media Center and Media Server closed then reopened to see/use the updated library.