How to get short definition of all built in Mathematica Functions?

Perhaps the simplest way is to use the built in function Information, which is the programmatic form of ??

Information[#, LongForm -> False] & /@ functionslist

gives a long list of (short) function definitions. By the way... it's easy to figure this kind of thing out -- in this case, I highlighted the symbol ?? (double question mark) and pressed the F1 key to bring up the help. That brought me to the function Information. I had first tried help on the single question mark (which brought me to the function Definition) which didn't seem quite right.

Here is the very start of the output...

You can also use the usage messages:

usageInfo = With[{func = Symbol[#]}, func::usage] &;

cF = Column[Style[#, 16, "Usage", Background -> None] & /@ #,
            Dividers -> All, Background -> {{LightBlue, LightOrange}}] &;

usageInfo /@ functionslist[[;;8]] // cF

Alternatively, make the usage message content a Hyperlink to the docs page:

usageInfo2 = With[{func = Symbol[#]}, 
    Hyperlink[Style[func::usage, 14, "Panel", Background -> None], 
     "paclet:ref/" <> ToString[func], ActiveStyle -> None]] &;

usageInfo2 /@ functionslist[[;; 8]] // 
       Column[#, Dividers -> All, Background -> {{LightBlue, LightOrange}}] &