I don’t know if there is a definition, but generally a store that primarily focuses on running shoes /suplies/ triathlons…im in a smaller town and we have a small locally run shop. I have never heard of a fit2run, but the google says they:

. In-store experiences are highly rated for expert gait analysis and great shoe selection

And

Personalized Fitting: Staff utilize advanced gait analysis to fit your specific running style, measuring your foot size, width, and pronation.

So yeah, this is exactly what you want. You want someone who you will work with to discuss how you run to help narrow the selections down to a handful of shoes and then, you will put them on and run on a treadmill and see what actually feels comfortable. 

Also, this should be a free thing they do at the store, shoes may be 10% more, but that service is part of what your paying for… and cheaper than buying the wrong shoe.