Newbie problem with transistor switching
This answer only uses BJTs, because you explicitly mentioned studying transistor theory.  An easier solution would be to use a comparator and a voltage reference.

Original schematic

I'll repeat your schematic to make it easier to read:

I cannot reproduce the 4.8V switch on.  Simulating on circuitlab shows a ~1.6V switch on (current through LED > 0.5mA).

Even if we fix the base biasing (to put \$V_{BE}\$ to ~0.6V), the switch on characteristic is quite soft.







simulate this circuit &ndash; Schematic created using CircuitLab

Second stage

By adding another stage, the switch characteristic is improved:







simulate this circuit