How does Storyline 360 identify whether it is published to an LMS or the web?

Maybe I'm confused here, but like Joe says, you have described creating and publishing two versions, one for the LMS, and a web version. If you have a true LMS, and not just a web server you are calling an LMS, it will take care of the problem. The function of an LMS is to store, deliver, and track eLearning.  The way your mockup describes it is that the learner will start using a web version (probably from a local file server), and if they have an account on the LMS, be asked to log in there (exiting from the web version), and start over with the version the LMS gives them to complete the training.

I'm not clear on what your are trying to accomplish, but I'm pretty sure you can trust the system: if they download it from the LMS, it is the LMS version, otherwise, it is the web version.  There really is no way a learner can start with a version opened from a web server, then go to the LMS to log in, and continue with the original version, and have the LMS track and report on them. The LMS will track and report only the version that it starts and sends out to them.