Use one or two transistors to switch an LED from PIC
The extra information does help.

The main difference between your 1st circuit and the right-hand lower circuit is how the load is connected.  Your circuit is known as a "High-Side Switch", the lower circuits are known as "Low-Side Switches".  The upper circuit also has more current gain than the lower circuits but that is important only when driving high-current loads.

Note that in both cases, the LED is turned ON when the input control signal is HI.

My preference is to use a Low-Side driver whenever possible.  It has the advantage of lower component cost.

The lower circuits that you included in your question describe what happens if you use a supply voltage that is too low for your LED.  A LED has a V-I curve where the LED conducts very little current below its' threshold voltage.  When the threshold voltage is reached, the current increases rapidly with very little increase in voltage across the LED.  

The left circuit is trying to show that: a 3.2V LED being fed from a 3.2V supply is just barely at the threshold point and doesn't draw much current.  It is therefore quite dim.

The right circuit shows the LED operating with normal current and shows you how to calculate the resistor value needed to set the current through the LED.