2-way design vs. 3-way design means ?

I am with Erik. And Richard. 

And I added crossover values to my list

https://speakerchoices.com/

The problem with questions like these is that not all 2-ways and 3-ways are created equal. There are high end 2 and 2.5-way speakers that can do everything better than 3-ways employing lower quality parts and materials.

Some 2-ways like the aforementioned Fritz speakers can sound more refined and play nearly as loud, or even louder, than many of the common $5000/pair 3-way from the likes of Revel, KEF and Paradigm.

IMHO, most 3-way designs, especially those under $10K/pair, don’t actually capitalize on the theoretical advantages of the design. Perhaps some of the newer value leaders from Arendal actually do, but then I wouldn’t be surprised if speakers like the 2-way JBL 4367s perform better subjectively.

The more experience I gain with DIY modding and building of speakers, the more I am becoming convinced that a speaker’s driver quality is what matters most. I would take a 2 or 2.5-way with some properly implemented Satori Textreme drivers over nearly any 3-way I’ve heard from the major manufacturers. 

A speaker like this will slaughter many/most 3-ways under $10K/pair:

https://www.madisoundspeakerstore.com/2-way-speaker-kits/satori-helios-textreme-9.5-2-way-speaker-kit-pair/

My understanding is that a 2 1/2 way runs the middle speaker full range (at least it does in the Wilson Benesch world). The woofer receives a low pass signal and the tweeter a high pass signal from the crossover, but the midrange is direct coupled to the amplifier and is not connected to the crossover. Theory being there is a purity of sound associated with eliminating the crossover component. The crossover points in my speaker are 5 kHz and 500 Hz, with the midrange acoustically rolling off to blend with the other drivers. 

No, not necessarily.

A 2-1/2 way just means that there are two woofers that are allowed to play all the way to their "bottoms" (i.e., no high-pass filter) and are low-passed at different frequencies.

My 2-1/2 way Mermans use a JBL 18" and a JBL 9.7" and neither uses a high-pass filter. The 18" crosses over at 310 Hz with a second order Butterworth and the 9.7" crosses over at 3,500 Hz to the Heil AMT, also with a second order Butterworth.  The 9.7 has a natural roll-off from ~300 Hz down and is rated at 400 watts RMS and I see no reason to "taint" it's sound by filtering it through a capacitor.

On my Super Big Reds, also a 2-1/2 way, the 15" Utahs have a slow roll off (first order?) from (IIRC) ~100 Hz and the 15" Altec 604E2 crosses over to the horn at 1,500 Hz.