Sentence #1 is clearly correct and natural.
I wouldn't say that #2 is grammatically incorrect (I can imagine contexts where it would be OK), but it does sound a bit off. I think the reason is that the habitual aspect of "I make" conflicts with the specificity of "a day".
To see this, let us put both sentences in the past tense, and consider:
3: I made ten phone calls a day.
4: I made ten phone calls in a day.
What do these mean? #3 talks about a habitual past action: "Back then, I was such a good salesman -- I made ten phone calls a day, and my boss always told people I was his most valuable employee."
In contrast, #4 talks about one past action on one particular day: "I was a pretty good salesman back then, but then this one day I was simply on fire: I made ten phone calls in a day!"
From this we can see why #2 doesn't quite work: "a day" here suggests a perfective aspect (one event which is completed), but "I make" is a habitual present. You could say "I can make 10 phone calls in a day", though.
(By the way, as a native speaker, your analysis feels totally natural to me, including your difficulty in putting your finger on how exactly to describe what #2 means -- I went through precisely the same thoughts.)