They're not really that different, actually. The one with parts is just split up into more folders (part folders), so you have to do a little work to get rid of the excess. First, move your chapter folders to the same level in the Binder as the part folders are now, then delete the (now empty) part folders. Now you should just have chapter folders and their children documents (scenes) in the Binder.
Next, go to the compile options and click "Formatting." You'll have two folder levels, Levels 1 and 2+. Level 1 is the Part folders and 2+ is the Chapter folders (and below, if you have any, but you probably don't). Click the 2+ folder and then the second button to the right of the "Options" button above (looks like a minus sign). This will leave you with one folder level (Level 1+, still technically a Part folder). Click it and then click the "Section Layout" button. In the "Prefix" box, change "Part" to "Chapter." If you want to use titles for your chapters, click the "Title" radio button next to the Level 1+ folder. This will append whatever you have your chapter folders titled to "Chapter I:", "Chapter II:", etc. Otherwise leave it unchecked and you'll just have Chapter I, Chapter II, etc. If you opt to not have the titles, go back into the "Section Layout" screen for the Level 1+ folder and delete the colon from the "Prefix" box, because that box won't know to get rid of it if you don't have a title after it.
Now, delete all document and document group levels to get down to Level 1+ for each of them. Click the "Section Layout" button for the documents and delete everything from the "Prefix" box, and uncheck the "Title" radio buttons for them. This will make it so the only title and "Chapter XX" prefix will occur at chapter breaks (determined by the location of your chapter folders). The scenes themselves won't be titled and will have whatever separator you choose to occur at text-to-text transitions.
This would give you a novel with this kind of formatting:
This scene one texty text is texty. Blah blah blah.
Scene two has even more text than scene one, yet it flows more smoothly.
Scene one of this chapter has the mention of dragons. But only the mention.
Etc. Adjust the settings to get it to look more like you want it.
I found that while tinkering around, waiting to hear back from someone, but I was too afraid to make changes. I'm assuming these changes stay with just this specific project and doesn't carry over to a new one or other projects I have? Is that a safe thing to assume?