In reply to Choi.

Different test statistics have different sampling distributions. In the link you provided, the section "Testing using Student's t-distribution" uses a test statistic that has a t distribution. The test statistic depends on the estimate of correlation AND an estimate of the standard error. In the section "Using the Fisher transformation," they use a different test statistic. This test statistic is approximately normal, hence a z score. The section also mentions that "In practice, ... hypothesis tests are usually carried out using the ... Fisher transformation," which is the topic of this blog post.