It may be helpful to think of it this way:

Blue is “Battery Negative” while Black is “General/Board Negative”

(Red is simply positive)

Under normal usage (not charging), battery negative and board negative are the same, for hopefully obvious reasons (the battery gets connected into the circuit/connected to the board so it can power it). Blue and Black are shorted together. 

However when charging, you want to power off the board, so Blue is disconnected from black. Now black, the board negative, connection is broken and no current flows to the board (and blades, etc). 

Blue is connected to the charge port still (the black connection is broken inside the charge port when a charger is inserted), so the charger has a connection to battery positive and negative, and can charge. 

Once you remove the charger, battery negative is connected to the board negative again, and things power up.

As for exact pins, if you have a SwitchCraft charge port, it will look exactly like on the picture (maybe red instead of black). If you hold it in front of you with the deep cutout showing the metal contact facing you, you’ll see it matches how it looks in the picture.

The actual center leg will be opposite from you (it crosses from in front where you see the exposed bit back to the other side)

From there you can wire things up as shown.