There is no right or wrong way.
The whole idea behind Mobile First is to start thinking about optimizing for the mobile experience first. Since the market share is shifting towards mobile devices you want to take advantage of this growth and shift as the industry pivots. There is more and more emphasis on mobile. So why not make that priority. And this is not necessarily related to how you “design”. Your design might start off as “Desktop First” but the way you build it out can be “Mobile First”. Design is just Design.
Think more about content delivery. Usually what happens with “Desktop First” is you’ll get… scroll… scroll… scroll… scroll… scroll… scroll… because you’re trying to fit everything on a small screen.
Presentation (part 1)
Presentation (part 2)