Hemi,
I wrote a reply, but StuntHanger ate it when I attached a photo that was too large for it to swallow....
Brett is correct that the purpose of getting the line/plane connection onto the bellcrank is to get rid of some drag. Also, many racers will have the lines coming out of the same hole I the wing, so connectors there would be problematic. Note that this also gets rid of one load-bearing line connection or joint, which if done right can only help the reliability.
There are a number of construction details that result in a practical setup. Rather that recreate my prior (lost) reply, let me know if you need more details.
Note that "buttons" actually may refer to two different kinds of setups. Button bellcranks are different than speed buttons, which are similar to scale racer internal connection buttons.... What Pat has shown are some of the details of a button 'crank, which was your original question, but understanding the others may eliminate the confusion you are having.
The first use of "button" was on the monoline control unit as far as I know. This would be the Victor Stanzel mechanism instead of the Jim Walker (bellcrank) mechanism. On the end of the Stanzel unit is a brass (or steel) part that looks pretty much like an old-fashioned shirt button (loose button, not sewed to the shirt). The control line loops over it and cinches down on it. Still used today in heavier sizes. These same buttons are used on internal 2-line connections on things like AMA Scale Racers and B-TRs where getting at the bellcrank is too much hassle to use a button bellcrank. If you want pics of any of these, send me a PM with a number and I'll text it to you.
Divot McSlow