R1 sets differential gain. If removed, the circuit will have gain set by R3, R4, R4, R5 (R3 and R4 in the second diagram). Benefit of R1 is that it increases differential gain while leaving common mode gain equal to 1 in both branches.
Another benefit of this topology is that an integrated instrumentation amplifier with high common mode rejection ratio can be made with accurately matching R2, R3, R4 resistors (easy on an IC) while their absolute value accuracy is not so important (not that easy on an IC) and set gain using only one external component.
Regarding naming, "non-inverting amplifier" is similar to "voltage follower". Former can have any positive (as opposed to negative, inverting) gain while later is typically unity gain: output follows input. In this circuit it is more accurate to call it "non-inverting amplifier" to emphasis the fact that both positive and negative input signals preserve their phase and are subtracted in A3. Though if R1 is removed, A1 and A2 becomes both "non-inverting amplifier" and special case of it - "voltage follower".
There is a Wikipedia article about Instrumentation amplifiers with lots of more good reading: