Fixed this myself, although still don't know how or why.
Used the steps in Doc ID 2237225.1 even though it was for a different problem:
A. Start adadmin:
B. Select 1. Generate Applications File menu
C. Select 4. Generate product JAR files (Note: Test with no force generation, however if JARs do not generate successfully the force option can also be tested)
D. Select 2. Generate form files (with library files; with menu files; with forms executables; for all products; all forms and all languages)
Logged in from internal network location and forms session launched.
Turns out it wasn't the adadmin that fixed it:
While adadmin was running, I had gone to the Java Control Panel in Windows, General tab > Temporary Internet files and deleted all the temp files that pertained to that specific instance (8070 in my case) and I was able to launch forms.
However, when I went to another PC, I got the same network error, so did the same Java temp file clearing and was able to launch successfully.
Sorry for the red herring, but all's well that ends well.