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I've been told many times that, when carrying an independent t-test, I have to determine whether my dependent variable (DP) is normally distributed on boths levels of my independent variable (IV).

On the other hand, I know that the independent t-test (and many other statistic test) is based on the General Linear Model (GLM):

Y = b0 + b1*X + e

Where Y is my DV (continuous) and X my IV (coded for example has -1 and +1). As far as I know, the GLM requires, among other things, that the errors are normally distributed. It doesn't say anything about the distribution of the IV or DV.

If my assumptions are correct, then why are we asked to check if our variables are normally distributed?

Also, assume I have a large sample.

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Both conditions are equivalent. Let us look at your equation $$ Y = b_0 + b_1X + \epsilon $$ with $\epsilon$ being normally distributed with mean zero.

Then samples in one group (indicated by $X=0$) are normally distributed with mean $b_0$ and samples in the other group (indicated by $X=1$) are normally distributed with mean $b_0 + b_1$. This is the same as the requirement for the t-test you cite above. What distribution $X$ has, or if it even has a distribution, is not relevant.

A normally distribured random variable plus some number is still normally distributed.

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