T-test table interpretation error?

For a research project I have obtained results which I have to test for significance. The two values I am comparing are the true diameter of a certain object versus the calculated diameter of that object. In this case it would be favourable if the t-test showed no significant difference between those results.

However while checking the T-table I came across this problem which just won't reach my brain. When my t-value is higher than a critical value there is a significant difference, but as the probability (alpha) increases that critical value increases. This leads to my misunderstanding that for this reason I can say that:

1. There is a 95% probability that there is a significant difference between the true values and calculated results

But since the critical value rises with probability:

1. There is a 99.9% probability that there is no significant difference between the two

This is obviously not right, but I cannot grasp what is going wrong.

When my t-value is higher than a critical value there is a significant difference, but as the probability (alpha) increases that critical value increases.

This is not the case. Conventionally, "alpha" refers to the significance level; when it gets smaller, the t critical value increases. You might be confusing alpha with its complement (1-alpha).

There is a 95% probability that there is a significant difference between the true values and calculated results

This is not a correct description.

You might say something like (if the null were true) "the probability of observing a t-value at least as large (in absolute size) as the one you obtained is less than 5%".

vs

"the probability of observing a t-value at least as large as the one you obtained is larger than 1%"

There's no inconsistency there.