The answer depends on the Enigma model, on the number of rotors among which the active rotors are chosen, on the number of wires used for the reflector, and on what one accounts for as part of a setting. The discrepancy between the two numbers around is because the position of the rotors (except the left one, which notch is inactive) can be accounted for - or not.
If we consider an Enigma M3 (as used by German forces at the beginning of WW2) with $r=3$ rotors chosen among $n=5$ rotors of wiring and notch positions known to the cryptanalyst; a plugboard with $w=10$ wires assumed all used; and define a setting as consisting of what's changeable, assumed unknown to the cryptanalyst, and remains constant during a session of use of the Enigma:
- The rotors can be ordered in ${n!\over(n-r)!}=60$ ways (Walzenlage).
- The relative position of the rotor and their internal wiring (Ringstellung) can be set to 26 values; for $26^r=17,576$ combinations.
- On the reflector, each of the $2w$ extremities of the wires can be plugged into any of 26 positions not otherwise occupied, with the two extremities of a given wire equivalent, as well as the wires themselves; for
${26!\over(26-2w)!\cdot2^w\cdot w!}=108,531,557,954,820,000$ reflector settings (Steckerverbindungen).
There are thus
$$s={n!\over(n-r)!}\cdot26^r\cdot{26!\over(26-2w)!\cdot2^w\cdot w!}=158,962,555,217,826,360,000$$
settings; that's $\approx1.6\cdot10^{20}\approx2^{67.1}$. This number is often quoted, and makes sense, as that's also the size of the key space according to the basic setup procedure; but this setup procedure has varied in secret ways, effectively increasing the key space.
That definition of setting does not account for the position of the $r$ rotors, which before actual encryption of a message is determined by the setup procedure, and is initially unknown to the cryptanalyst. Accounting for that, there are thus $26^r s$ states of the Enigma hardware when encryption begins.
For the left rotor, the position of the outer of the rotor is immaterial to encryption and decryption. That thus leaves $26^{r-1} s$ plaintext/ciphertext transformations implemented by the Enigma and distinguishable for long-enough messages; that's $\approx1.1\cdot10^{23}\approx2^{76.5}$. This agrees with Graham Ellsbury's Complexity of the Enigma: if a cryptanalyst is able to fully decipher a sizable message sent using an entirely unknown setup procedure, s/he has in effect recovered about $76.5$ bit worth of keying material.
Caution: none of the above numbers is related to the much lower number of operations involved by the attacks actually carried for cryptanalysis during WW2, which exploited weaknesses in both the transformation performed by the Enigma, and its setup procedures.
Note: some parameters and hypothesis used in the above computations are debatable:
- Ultimately, $n=8$ rotors where used for the Naval Enigma according to that source.
- Enigma M4 (Naval version after February 1942) had $r=4$ rotors; but these where not fully interchangeable, so the formula given does not apply.
- I can't tell if the number of wires used in the Steckerverbindungen was always exactly $w=10$, or variable in some range, or/and known to the cryptanalyst.
