Preventing duplication of code; parametrize a single function, or create two separate functions?

I'm in the process of combing through and simplifying a codebase for a python project. The nature of the selenium-wrapping library that I'm using called Splinter is that I find myself writing a lot of code with minor differences, maybe in the element I'm searching for, or a tiny change in selection logic, different parameters, etc.

Currently, I might have two separate functions with 90% of the code copy and pasted between them. My two ideas for condensing this and being consistent are as follows:

1) Create three functions: A, B, and C. Functions A and B would be called directly and have single parameters. These functions then call function C with both the parameter they were given, and then the correct 'identifying' parameter, to change the way that function C works. Function C would never be called directly.

def A(x):
        return C(x, 0)
    def B(y):
        return C(y, 1)
    def C(a, b):
        if b:
        else:

2) Create one function, and have it take two parameters. First parameter is what you would otherwise pass into A or B, and the second parameter is the 'identifier' portion.

def D(x,i):
    if i == 'case 1':
    elif i == 'case 2':
    else:

The 'identifier' part is what I don't like. Is it a smell to have a function depend on the caller using specific keywords, or values, in a parameter?

Option 1 produces two more functions to take care of, but option 2 causes a user to know these special 'ID' values in order for the function call to work correctly.

Which of these two patterns would be a better solution in this case?