Voltage Drop Across Diode in Off State
They are probably trying to show you the voltage the LED would drop when some current is going thru it.  I agree that is inconsistent with how they are showing the voltages across other components.  In reality, the LED will have very close to 0 V across it when the transistor is off.  The only current will be the transistor leakage, which quite small.

There is a case where what they show could actually be true.  LEDs also work as photocells in reverse, although rather poorly.  With no load on it and in reasonable light, the LED will develop a voltage close to its normal forward operating voltage.  However, the impedance of that will be so high that even a ordinary voltmeter can load it.  Whoever made that diagram may have probed around the circuit with a voltmeter, and at that illumination and that voltmeter, that's what was reported across the LED.  Whatever part of the supply voltage that doesn't appear across the LED would then be across the transistors, since no current is flowing.

Another possibility is that when they probed across the transistor with a voltmeter, the meter caused enough current to flow for the LED to develop 1.3 V across it, so the meter read 3.7 V.  They then subtracted 5 V from 3.7 V to say the LED had 1.3 V across it.  If the LED was measured directly, it would have less voltage across it.