Sorry, i pretty much skipped the following replies since then.
The point of executing commands manually is to try getting errors that would hint at what the issue is.
From what i read, JNLP is part of Java Web Start, which has been deprecated by Oracle, though IcedTea-Web may continue to implement it. If you do have IcedTed-Web in both machines, the different might be the version installed is different, the second one not supporting JNLP anymore.
There may be a way to manually “run” a JNLP from the terminal, but i don’t know the relevant options. You’ll need to read its manual to do so.
Also, as @linux-aarhus pointed, you may be able to receive support from the commercial editor. Even if they don’t update anymore, as long as they keep selling it, they should still be required to provide support for their products.
Or, considering the feature is deprecated upstream, you could find a different way to download the application.