White to play  

influences the crosscut. Suppose development towards the bottom right is important.

 

With this atari and connect, White limits the development of the Black stone and influences the bottom right.

At the same time, is weakened, and may possibly be captured in a ladder ().

 

Of course, Black can thwart White's intentions, but now comes out with a ponnuki and probably sente, thanks to having given atari.

 

Following the proverb and extend in this situation is not bad, but then Black can choose to connect to . Black's shape is not bad, whereas looks superfluous now.

 

Which is why is usually played when White can follow up with or a.

 

here violates the spirit of the proverb as it doesn't gain liberties. Now is perfectly placed as hane at the head.

From joseki (3-4 point distant low approach, one-space low pincer)  

Black plays atari and connect twice to damage .

 

Extend or atari?

Atari, atari  

A GoBase search indicates that this (or its transposition) is a popular pro continuation. ( may be a bit overconcentrated.)

Extend  

But is also common.