T value vs T-stat

When you assume that your test statistic (or loosely speaking the signal/noise ratio) follows a standard normal distribution, you could calculate z-scores instead of t-values. This could be the case if you sampled the entire population, or when you have access to a very large sample size.

On the other hand, when your sample size is small and hence the expected uncertainty in your estimates is likely larger, the t-distribution is more appropriate because it allows for more probability in the tails (fatter tails) when calculating the p-values from it. With a sample size of larger than 30, the t-distribution looks very much like the standard normal distribution (https://en.wikipedia.org/wiki/Student%27s_t-distribution#Definition).

As for t-value and t-statistic, I'd don't see a problem to use them interchangeably. The sample mean is also called a statistic but can also be labeled a value. What's important is that sample statistics (sample mean, sample standard deviation, t-value, z-scores) allow you to make inferences about the underlying population parameters.

Note however, that there is also something called the T-Score, which should not be confused with the t-statistic or t-value. From Wikipedia (https://en.wikipedia.org/wiki/Standard_score#T-score)

In educational assessment, T-score is a standard score Z shifted and scaled to have a mean of 50 and a standard deviation of 10. In bone density measurements, the T-score is the standard score of the measurement compared to the population of healthy 30-year-old adults.