I'm not completely sure about equivalence of two definitions of Turing machine.
The first one states that Turing machine has a finite alphabet $\Sigma$, set of states and some rules.
Turing machine in the second definition has two alphabets - input one $\Sigma_1$ and working one $\Sigma_2$ and set of states and rules.
I think it doesn't matter which definition you use, but I didn't find any proof or at least some mention about it anywhere.
The biggest "problem" I have is when you prove that multi-tape Turing machine is equivalent to one-tape Turing machine because in the proof you extend the alphabet (for example here). When you use the second definition you extend just the working alphabet and everything is ok. When you have the first definition you have to extend the "input" alphabet so the modified one-tape Turing machine accept more languages than the original multi-tape one and they are not equivalent...
Do you know where this problem (equivalence of the two definitions) is discussed? Or can you explain me that it does not matter?