Simply add a @ to the on $:TEXT: event so it looks like this -> @$:TEXT:

on @$*:TEXT:/^[!@.]define( |$)/iS:*:{
  if (!$($+(%,-,$site,.,$cid,.,$nick),2)) && ($regex($nick($chan,$nick).pnick,/(!|~|&|@)/)) { inc -u3 $+(%,-,$site,.,$cid,.,$nick) | var %ud = $+(ud,$str($ticks,3))
  $+(sock,$iif($sock(%ud),close,open) %ud api.urbandictionary.com 80) | sockmark %ud $replace($strip($2-),$chr(32),+) sockwrite -nt %ud .msg $iif(#,#,$nick) }
}

Above will only work if the user is OP'eD (@)
NOTE: Have "only" added the text event here... you'll need the rest of the script for it to work.

EDiT
I might have misunderstood you... the above will only work if the bot / host ( client running the script ) is OP'eD (@) and come to think of it that wasn't what you asked for I'm guessing ?

on *:sockopen:ud*:{ tokenize 32 $sock($sockname).mark 
  if ($sockerr) { $5-6 Socket Error! | halt }
  $2-4 GET $+(/v0/define?term=,$1) HTTP/1.0
  $2-4 Host: $+($sock($sockname).addr,$str($crlf,2)) 
  $2-4 Connection: close
}
on $*:TEXT:/^[!@.]define( |$)/iS:*:{
  if (!$($+(%,-,$site,.,$cid,.,$nick),2)) && ($regex($nick($chan,$nick).pnick,/(!|~|&|@)/)) { inc -u3 $+(%,-,$site,.,$cid,.,$nick) | var %ud = $+(ud,$str($ticks,3))
  $+(sock,$iif($sock(%ud),close,open) %ud api.urbandictionary.com 80) | sockmark %ud $replace($strip($2-),$chr(32),+) sockwrite -nt %ud .msg $iif(#,#,$nick) }
}
on *:sockread:ud*:{ sockread &ud
  tokenize 32 $sock($sockname).mark
  if ($sockerr) { $5-6 Socket Error! }
  elseif (no_results isin $bvar(&ud,1-).text) { $5-6 $qt($1) is undefined! }
  elseif ($regex($bvar(&ud,1-).text,/"definition":"(.*)/)) { tokenize 34 $regml(1) | $gettok($sock($sockname).mark,5-6,32) $regsubex($1,/\Q\r\n\E/g,$chr(32)) }
  ;*** Use the one below and comment the one above out if you only want 1 line ( sockclose added to the one below )
  elseif ($regex($bvar(&ud,1-).text,/"definition":"(.*)/)) { tokenize 34 $regml(1) | $gettok($sock($sockname).mark,5-6,32) $regsubex($1,/\Q\r\n\E/g,$chr(32)) | sockclose $sockname }
}

The above will "only" work if the user is OP'eD (@) or above
And have added a extra line with sockclose in case people only want the first definition.