Your assumptions are partially correct. The wiring of the rotors is technically part of the algorithm, but because of the era in which Enigma was invented the rotors were kept secret, so discovering the rotor wiring became an initial step in the cryptanalysis of the machine. The plugboard/steckerboard, however, was changed by the operator every day and its setting was definitely part of the key.
The full key to an Enigma message is made up of everything the operator needed to set in order to encipher and decipher a message. There are four parts to the key.
- Rotor order. Each rotor had a Roman numeral engraved on it to identify it, and the operator had to install them in the assigned position. This was set daily.
- Ring position. Each rotor had an alphabet ring that could rotate about the rotor, and it had to be positioned correctly before installing the rotor in the machine. This was set daily.
- Letter starting position. Each rotor had to be rotated to display the correct starting letter. This was set at the beginning of each message.
- Plugboard wiring. This was set daily.
The settings were changed every day. The operator looked up the settings by date printed on a sheet of paper and would set up the machine for that day, establishing the key. The starting position of the wheels had to be set prior to each message being sent or received. If there was a problem or typo during the message, the operator had to start over by resetting the wheels, and retyping the entire message.
I believe the setting sheets were distributed monthly, and the naval versions were printed on water soluble paper so they could be destroyed in the event of imminent capture or sinking; but I don't have a reference for that.
Operating instructions for the machine can be viewed here: http://www.ellsbury.com/enigma3.htm
An English translation of a manual for the machine can be found here: