Let me add something about language. Go has been around for a long time, and a lot of go terms are like words in regular language. They have more than one meaning, which are not always precise. Here there are two distinct meanings of "connected".
The first meaning is the basic one, where stones of the same color may be connected along the lines of the board.
$$Bc
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . O B B . . .. .
$$ | . . O O B . X . .
$$ | . . . . . . . . .
$$-------------------
- Click Here To Show Diagram Code
[go]$$Bc
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . O B B . . .. .
$$ | . . O O B . X . .
$$ | . . . . . . . . .
$$-------------------[/go]
The marked Black stones are connected in that sense. I think that that is clear to you.
There is another sense of
connected that is a little hard to explain, and different people explain it differently. In that sense the other Black stone is connected to the marked stones.
Suppose that we are playing a go like game where Black wins if she can connect all four Black stones in the first, basic sense, even if White plays first. If Black can win that game, then the Black stones are connected in the second sense.
As you already know, the game might go this way:
$$Wc Black is connected
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . O X X 2 3 . .
$$ | . . O O X 1 X 5 .
$$ | . . . . . 4 . . .
$$-------------------
- Click Here To Show Diagram Code
[go]$$Wc Black is connected
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . O X X 2 3 . .
$$ | . . O O X 1 X 5 .
$$ | . . . . . 4 . . .
$$-------------------[/go]

at 1.
With

Black connects in the first sense, and wins the game, so Black is connected in the second sense.

Now, we may also consider the Black stones to be connected after

, even without

, because for White to play at 1 would be illegal. That is a third sense of
connected.
Most people do not think about these things in so formal a manner, they just picked up the idea as they learned the game.
