Which sounds better 2 way or 3 way speaker design

I am under the impression, (which I may be incorrect) that a three way design is superior to a 2 way design.  
 

Wrong - it’s the quality of the parts and implementation.  Some crossovers are done extremely well they seem seamless.   More crossovers “may” offer more opportunities to have a poor crossover, but it’s not automatic.

As usual, best to demo yourself to hear what resonates with you.

Ime loudspeaker design is a fascinating competition of ideas, every one of which is a juggling of tradeoffs (and anyone who tells you otherwise is in marketing). The implementation of those ideas can take very different forms: Single driver, two-way, three-way, four-way, planars, horns, multidirectionals, omnis... And even among speakers which are outwardly similar, the designers are constrained to giving some attributes higher priority than others.

In general the attributes which I prioritize are more likely to be found in a conventional three-way than in a conventional two-way, but my own designs tend to be unconventional two-ways with rear-firing tweeters, often augmented by subs... in which case I guess they are three-way systems?

Duke

One of my favorite basic design is a three-way, with a compression-driver/horn midrange covering a wide part of the range.  The compression drivers I like are all vintage—from 1930’s to 1960’s.  I own a system fitting that description—twin 12” woofers (modern), bullet tweeter (modern), western Electric 713b midrange compression drivers (1940’s), Western Electric KS 12025 multicellular horn.

I mentioned two single driver speaker companies I like-Charney and Songer—above, and I would now like to give kudos to PureAudioProject which makes terrific open-baffle speakers, modt of them are two-way speakers.  

There are good speakers of all types.
 

Having owned Proac speakers in the past I would not rule out the K3 without having an audition. Unfortunately my listening room will not accommodate them but .....

A great discussion.  I’ve had so many speakers over the years from single element to 4-way in many different rooms including large towers and smaller bookshelves.  I agree with the comments that stress design and execution including component and build quality.   My old friend that was involved in speaker design and build always said there is no perfect speaker and that there were trade-offs in any design and build.  I find a key is the room you are listening in and what sounds good to you at this time.  I now listen in a relatively small room so small towers or 2-way bookshelves fill the room better without overpowering the space.  In addition my old man ears don’t hear as high in range as I used to so sibilance is key as as such there are speakers I liked when I was younger that I wouldn’t listen to now.  Trust your ears and what sounds good to you in your listening room.  I haven’t had those 3 speakers but With an 18K budget you’ve got some great choices.