The usual Student's t distribribution has only one parameter, known as the degrees of freedom.
Please see the link.
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The mean and standard deviation.
The Student's T- Distribution is a type of probability distribution that is theoretical and resembles a normal distribution. The Student T- Distribution differs from the normal distribution by its degrees of freedom.
No they are not the same.
z-scores are distributed according to the standard normal distribution. That is, with the parameters: mean 0 and variance 1.
Check the lecture on t distributions at StatLect. It is explained there.