After comparing some text books and lectures about computability theory I encountered not only the halting problem $$H = \{\langle M,x \rangle \mid \text{the Turing machine $M$ halts on input $x$}\}$$ but also the blank-tape halting problem $$H_0 = \{\langle M \rangle \mid \text{the Turing machine $M$ halts on input $\varepsilon$}\}.$$ Here, $\varepsilon$ denotes the empty word.
I know that $H_0$ is as undecidable as $H$ is, since they can both be reduced to each other, but I also know that any reduction from $H_0$ (e.g. the one from the proof of Rice's theorem) could also be done from $H$. What intrigues me is why so many authors first introduce $H_0$ and then use it in their reductions instead of just using $H$. I do not think the proofs become significantly simpler.
My guess is that it has an historical reason but I cannot find where $H_0$ was introduced the first time.