Java is a bit of a mess. (Ok a lot of mess. :) )
Some software now comes with an embedded JRE to run Java code by itself.
Some software - even allegedly current software has been written to specifically require ye olde Apple Java 1.6
Some software requires a JDK version of Java rather than a JRE but this is typically only for server type software or develop tools.
It sounds like your Logitech software is still wanting the ancient and venerable Apple Java 1.6.
Apple have been warning for years that they were going to discontinue this old version and according to some definitions have discontinued it in that they are not updating it to fix bugs. They did however under pressure update it a number of years ago so it would work with newer versions of macOS and this includes Mojave but have announced that this time definitely, really, no chance they will change their minds again not be updating it to work in macOS Catalina.
Therefore those foolish developers who have stubbornly and stupidly failed to update their software will find their programs, dead, dead, dead very soon.
On top of the above Oracle who are the 'owners' of Java have changed the rules.
It used to be that using Java was fairly straight forward. Now unless it is for personal use or purely development you need a license. You also need to create a user account to download updates. They have also changed to rather than infrequently issuing massive updates to more frequently issuing smaller updates. This means the version number is now increasing at a much more rapid rate. Remember Apple's version that is arguable version 6, for a long time the most common version was version 8 we are now on version 13 with version 14 already in development.
Due to the licensing changes by Oracle and the fact Oracle is not considered pleasant to deal with as a company, there is now an open source version of Java called OpenJDK - it is aimed more at developers. There is also another version called Adopt OpenJDK.
A JDK includes the JRE but a JRE does not include the JDK. (If you have to ask you don't need to know.)
Another forthcoming change is that Apple are removing (trying to kill off) scripting languages like Python, Perl, PHP in macOS Catalina. This strictly speaking does not affect Java which already has to be manually installed but it might indicate changes will make Java 'harder' to get working.