Which sounds better 2 way or 3 way speaker design

It depends on the design. There are pros and cons with each, but there are a lot of outstanding 2-way speakers.  They're simpler and can typically offer higher quality parts because there’s simply fewer of them to buy. Because of they can be built for less, many are indeed marketed to the more affordable price ranges, but a TOTL 2-way can still be amazing, depending on the recipe used.  

With that said, there are a lot of great sounding 3-way + speakers out there too. Keep an open mind, buy what you like, and what works in your room on your system, and you won’t go wrong.

"Dedicated midrange driver" is typically an oxymoron and very few three-way systems will not put some of the midrange in the woofer and/or tweeter.  You would need to run the driver from ~80Hz-~4kHz for it truly to be "dedicated" to the midrange.

@toddalin , I always am interested in your perspective.  After reading your above post, how would you answer the OP's question?  

Also, if one was  going to go with a two way speaker, is there a sonic advantage in a two way that is built into its own floor standing cabinet versus a two way that is intended to be placed on a speaker stand?

I prefer 2 way stand-mounts with specific OE stands that are designed for use with the specific 2 way transducer. 

I have 2.5 way floor standers and they sound nice bust not as clear, detailed or have the depth or breadth of my 2 ways. 

Now there are many variables that play into this also, room size, room treatments and so forth. 

I have no knowledge of the three speakers in question.

Whether a 2 or 3 (or 2.5 or other) way is best is dependent on the selected components, construction, and it’s implementation. There are fine speakers made in many "ways" and I don’t think that the actual number of drivers is the important factor in deciding which to choose.

In general, I would think that a floor standing speaker would offer more cabinet volume and this would typically allow for a deeper, more generous bass. But of course there are exceptions to everything.