Okay I get it. The newly translated page is automatically 'checked' as being the default page... which is why it's path is set to "". The other pages will have a path of pagename + culture.
That default page is what is shown when going to www.mywebsite.com. The other pages can be navigated to with path. i.e. www.mywebsite.com/home-fr-FR.
The trick is to manually set the page that matches the default culture to be the default page. That way, it will be the at www.mywebsite.com and the others can be navigated to with the path as needed.
I'm not sure if it would be more SEO friendly to have the home page translations use the culture as the path i.e. www.mywebsite.com/fr-FR ? I am not an SEO expert so I'll leave it as something to concider.