How does the ring settings of enigma change wiring tables?
There's two missing pieces.  First, the ring setting changes the output letter, it doesn't rotate the whole exit pattern.  Second, the rotors are advanced before the letter is encrypted.

If your rotor (Enigma I Rotor I) is set up like this with the ring at A.

abcdefghijklmnopqrstuvwxyz
ekmflgdqvzntowyhxuspaibrcj


Then if you advance the ring to B all the outputs will be shifted up the alphabet by one.  A will become F instead of E.  B will become L instead of K.  And so on.

abcdefghijklmnopqrstuvwxyz
flngmherwaoupxziyvtqbjcsdk


This almost matches the simulator, but why is the simulator's output shifted over one?

abcdefghijklmnopqrstuvwxyz
kflngmherwaoupxziyvtqbjcsd



  Remember that, when depressing a key, the rotors advance before the electrical signal runs through the rotors. Therefore, to examine the current flow through the rightmost rotor in A position, the rotor must be set in the Z position before depressing the key (this also counts for the other rotors if they are due to step).


Source: Technical Details of the Enigma Machine

When you press A the rotor advances to B and encrypts the letter according to B.  Your simulator has to first advance the rotor and then encrypt the letter.