When illustrating what states are in Turing machine, often the examples of programs, like a checker that checks an input number is even number, are given. But different programs seem to have different number of states - if so, how can Turing machine be described with the set of finite number of states, along with others? In other words, if we follow the formal definition of Turing machine, can we construct a state diagram of a Turing diagram, assuming infinite memory exists?
There are two possible meanings of state here:
One of the finite states in the finite control mechanism of the Turing machine.
The entire state of the Turing machine, including its internal state (the one from the previous definition), the locations of the heads, and the contents of the tapes.
If you are looking for a state diagram of the second type, then it's infinite. If you're looking for a state diagram of the first type, then it's finite.
Different programs seem to have different number of states – every program is finite (has finitely many states, under the first interpretation above), but there is no bound on its length: it can be as long as you want, as long as it's finite. The same is true for Turing machines.