Your link did not work for me. As you know the frequency and the tuning capacitance you can work out the inductance of the primary. If you connected the transistor directly to the transformer with the other end going to Vcc, the output impedance of the transistor would damp the tuned circuit , resulting in excessive bandwidth. The collector is taken to a tap on the primary to reduce this effect at the expense of a slight increase of losses in the tune circuit. The base of the next transistor is again a much lower impedance, so between it and the number of turns for the collector circuit, there is a step down ratio of about 4 :1. So one scenario could be primary 200 turns, tapped at 80 for the collector and Vcc, base secondary 20 turns. wire gauge not importent, though primary should be wound with Litz wire. The actual number of turns depends on the ferrite core (or not). The reason for different types is that one could be for the mixer/oscillator, first If amp, second IF amp and detector, all requiring different impedance matching.
Frank