Q:

Why do I get a different result when calculating the probability in a t distribution using a Stat Trek calculator?

From Online Stat Book:

I used the Stat Trek t-distribution calculator instead, and got this:

Why is the result different from the one provided in the test, and the one calculated using the Online Stat Book t-distibution calculator? 

A:

This question you have pasted in concerns me a lot.
The premise of the question is wrong.
The answer to the question "In a t distribution with 10 degrees of freedom, what is the probability of getting a value within 2 standard deviations of the mean?" is NOT obtained by finding the area under the t-density between -2 and 2.
Unlike the case with a standard normal distribution, the standard deviation of a "standard" t-distribution (i.e. the one in t-tables) is never 1 at any finite degrees of freedom.  With $10$ df this standard deviation is $\sqrt{10/8}\approx 1.118$. So the probability of getting a value within 2 standard deviations of the mean in a t-distribution with 10df is the area under the density between -2.236 and +2.236 (approximately ... and the answer to that is 0.9507)
If the question intends to ask something for which the area between -2 and 2 is the answer, it either has to give those values specifically (like "find the probability that a t-distributed variable lies between -2 and 2"), or it has to refer to the standard deviation of a different variable than the one with the t-distribution (such as the variable on which a t-statistic is calculated).

To respond to the issue of why the two values are different, you can see in the output of the Stat-Trek Calculator that it gives $P(T\leq 2)$, not $P(-2\leq T \leq 2)$ -- it's right there in the last line of your screenshot. [It also calls it "cumulative probability", which is a second way of indicating it's calculating the area to the left of its argument]

