Input pin and output pin have the same naming convention '1' or 'pin1' etc, whether input pin or output pin depends on the context of the command used - SERIN 1 is an input command so references an input pin, SEROUT 1 is an output command so references an output pin. For use of "pin1", again that's context "pin1 = b0" sets an output pn, "= pin1" reads an input pin, likewise with "If pin1 = 1 Then".
"Pin" is the name given to a PICAXE connection in or out, their location on each chip is defined in PICAXE Manual 1. To differentiate between the leg number of the chip ( also called "pin" by the industry ) we have generally adopted the terminology "leg" for referring to a physical chip leg, "pin" when referring to a software pin.
Some people disagree with this "leg" and "pin" terminology but IMO is a handy and convenient shortform; "output pin 1 is Leg 21 of a 20M" against "software output pin 1 is physical chip pin 21 of a 20M", and the temptation to shorten the later to "pin 1 is on pin 21", which is quite confusing.