My professor said that "You should never accept the null hypothesis, instead you state there is insufficient evidence to reject the null hypothesis" and that led me to a loophole of semantics.
Why is it not appropriate to "accept the null hypothesis", and does rejecting the null hypothesis imply that you "accept the alternative hypothesis"?
Furthermore what would be the most correct way of stating your result if the test statistic were to lie in the critical region?
I take the question to arise from traditional frequentist hypothesis testing, not from a Bayesian context.
If you are at the beginning of the theory of hypothesis testing, as with the Neyman-Pearson fundamental lemma, you will likely have a simple hypothesis and a simple alternative such as $H_0: \mu = 100$ against $H_a:\mu = 110.$ Then it is common to say you accept either $H_0$ or $H_a.$
In elementary courses, it is common to use terminology such as "fail to reject" $H_0$ or "retain" $H_0,$ but to say that you "reject" $H_a.$ The goal of this seems to be to stress that $H_0$ and $H_a$ are often not on an equal footing.
The equal sign in $H_0$ often specifies a particular parameter value, leading to a specific 'null' distribution, which is used to compute a P-value from the observed data. If the P-value is quite small you can say it would be rare to get the observed data if $H_0$ were true, which might lead you to reject $H_0.$ By contrast, if the observed data would not be unusual if $H_0$ were true, that gives you no strong probability statement leading to 'accept' $H_0$, which might be twisted to mean 'wholeheartedly embrace' $H_0.$
Often the 'purpose' of an experiment is to see if there is evidence that $H_0$ is wrong and and so ought to be 'rejected'. [If this experiment doesn't sink the null hypothesis, maybe the next one will.] But in many instances (especially with goodness-of-fit tests) the goal may be to establish that $H_0$ is, at least, not unreasonable.
Even so, it may feel strange to say 'fail to reject' $H_0$, but to have no qualms about saying 'accept' $H_a.$
In some of my classes I have made a surreptitious deal with my students that I will try to avoid the perhaps triple negative phrase "fail to reject the null hypothesis." (Null hypothesis itself may often be taken as a negative statement). Instead, I may sometimes say "accept" $H_0,$ if they will promise not to mis-construe that as a full endorsement. (Reminders are made periodically that this is considered forbidden language by some, perhaps not quite as bad as the banned four letter words.)
With this background and in the privacy of your own room with nobody within ear-shot, you have my permission to mutter quietly "accept the null hypothesis." And without guilt.
Perhaps the most useful and important thing is to remember that the proper result of a frequentist significance test is to make a statement about the data. The data are a bad match for the null hypothesis or the data do not seem to cast much doubt on it.