In quantum error correction, two stabilizer codes are considered to be equivalent if they differ by a local Clifford transformation or a qubit permutation. Represented as a tableau, these set of transformations establish an equivalence class between different tableaus that represent equivalent codes.
The literature suggests that, using this notion of code equivalence, there is a unique $[\![5,1,3]\!]$ code with 4 cyclic stabilizer generators $g_1 = XZZX1,~g_2 = 1XZZX, \dots $ and logical basis $X_L = XXXXX$, $Z_L = ZZZZZ$. I have no problem in generating a tableau for such a representation of the code, and I can also verify that it is indeed a $d=3$ code by checking that the (Knill-Laflamme) error-correcting conditions are satisfied for a standard symmetric depolarizing channel.
However, I have also been able to find a $[\![5,1,3]\!]$ code (i.e. I can satisfy the same Knill-Laflamme equations as in the previous code) represented by a tableau in which both $X_L$ and $Z_L$ are weight 3 Pauli strings. To me it doesn't seem that such a code can be mapped to the standard $[\![5,1,3]\!]$ code that I've described in the previous paragraph using only local Clifford transformations. The reason is that I don't see a way in which the weight of the logical operators can be changed using these transformations.
My question(s) are thus the following:
In which sense is the $[\![5,1,3]\!]$ code unique?
Is it true that two different tableaus can only represent equivalent codes if all the weights of their Pauli strings are the same?
I'm aware of https://physics.stackexchange.com/questions/283008/equivalent-quantum-codes but it doesn't answer my question.