Calculate the Numpad Taxicab

Jelly, 11 bytes

o-1.’d3ạ/ḞS

Try it online! or verify all test cases.

How it works

o-1.’d3ạ/ḞS  Main link. Argument: [a, b] (list of integers)

 -1.         Yield -1.5.
o            Take the logical OR of a and b with -1.5.
             This maps 0 to -1.5, and all other integers to themselves.
    ’        Decrement the results.
     d3      Divmod; compute quotient and remainder of each result divided by 3.
       ạ/    Reduce by absolute difference across columns.
         Ḟ   Floor; round all differences down to the nearest integer.
          S  Sum; add the rounded absolute differences.

Python, 140 114 bytes

I am a first timer, so please help. Here's my code.

def f(x,y):x,y,k=x-1,y-1,abs;return x!=-1 and y!=-1 and k(y//3-x//3)+k(y%3-x%3) or k(y//3-x//3)+int(max(x,y)%3==2)

Jelly, 13 bytes

+2$.¹?€d3ạ/SḞ

Try it online!

Port of my Pyth answer.

Test suite.