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No, the Z-test is not the same as a Z-score. The Z-test is where you take the Z-score and compare it to a critical value to determine if the null hypothesis will be rejected or fail to be rejected.
The Z-score is just the score. The Z-test uses the Z-score to compare to the critical value. That is then used to establish if the null hypothesis is refused.
a t test is used inplace of a z-test when the population standard deviation is unknown.
z test
The z-score, for a value z, is the probability that a Standard Normal random variable will have a value greater than z.