Have a look at this article:
https://en.wikipedia.org/wiki/Two-port_network
Under the heading "Admittance parameters" you will see a 2x2 matrix with elements y11, y12, y21, y22. The elements on the main diagonal (y11 and y22) are the "driving point admittances" (conductances) at the input and output of the two-port. They allow you to calculate the ratio of current to voltage (that ratio is an admittance or a conductance) at a point in a circuit when you apply a signal to that point.
The off-diagonal elements (y12 and y21 in the case of a two-port; a more complicated circuit could have a larger admittance matrix with more than 4 elements) are the "transfer admittances". They allow you to calculate the current in a part of the circuit which is different from the part where you apply the signal (voltage). The word "transfer" is used because those elements of the matrix relate the voltage in one part of the circuit to the current in another part of the circuit.
When analyzing vacuum tube circuits, the tube can be treated as a two-port, and the admittance parameters are appropriate to use because the input impedance of a vacuum tube is extremely high at low frequencies. The y21 parameter is the important parameter that relates the current in the plate circuit to the voltage applied to the grid. It's full name is "forward transfer admittance" (or "forward transfer conductance" at low frequencies). The name is commonly abbreviated to "transconductance".
When John Pierce coined the name "transistor", the transistors available at the time were mostly point contact transistors. Their input impedance was much lower than that of a vacuum tube, and it was more appropriate to use impedance parameters rather than admittance parameters to analyze transistor circuits.
Look under the heading "Impedance parameters" in:
https://en.wikipedia.org/wiki/Two-port_network.
The parameters z11 and z22 are the "driving point impedances". Just as in the admittance parameter description of a two-port, the z12 and z21 parameters are transfer parameters, and they describe the relation between a current in one part of a network and the voltage in another part, hence the word "transfer". The important parameter in this case is z21, which accounts for the property of providing gain. The full name of this parameter is "forward transfer impedance" (or "forward transfer resistance" at low frequencies). It is usually abbreviated "transresistance". That's what Pierce was thinking of when he coined the word "transistor".