what difference does slow cooking onions with/without broth make?
The issue is caramelization-- by cooking onions fairly slowly for a long period of time you can bring out the natural sweetness of the onions and caramelize the natural sugars present in the vegetable.  Onion soup (assuming you mean a French onion soup kind of onion soup) starts with caramelized onions and then adds broth, simmers, tops with bread/croutons and cheese, and bakes til bubbly and browned for a delicious soup that is rich, lightly sweet, and sometimes a little salty.

Making onion soup in a slow cooker is usually done for convenience, as the process of caramelizing the onions can be time consuming and requires frequent stirring (so it isn't idle time as with a crock pot or slow cooker).  This process is important for the richer, sweeter flavor it imparts to the soup.

The camp that adds onions alone to the pot and cooks for a while before adding broth is attempting to caramelize the onions, albeit in a more hands-off way, which would give a soup that is very close to the traditional method.  

The recipes that simply add raw onions and broth immediately are forgoing the depth of flavor provided by caramelizing the onions first-- unless they have indicated that the onions should be caramelized before being added to the slow cooker, there's just no way to replicate that flavor, so soups that start this way will have a less developed (not necessarily weaker-- but probably closer to raw onion than sweet caramelized onion) onion flavor.  

Personally, I find the broth-later versions to give a better result, richer and with more flavor.