If the natural density (relative to the primes) exists, then the Dirichlet density also exists, and the two are equal
UPDATED: As Franz suspected, `es steht schon bei Landau.'
On p. 118 of the first volume of Landau's Handbuch, one finds the following theorem: Let $f(s)=\sum_{n\ge 1} a_n/n^s$ be a Dirichlet series (with real coefficients $a_n$) that converges for $s>1$. Let $S(x)=\sum_{n \le x} a_n$. Then $$\limsup_{s\downarrow 1} \frac{f(s)}{\log\frac{1}{s-1}} \le \limsup_{x\to\infty} \frac{S(x)}{x/\log{x}},$$ and $$\liminf_{s\downarrow 1} \frac{f(s)}{\log\frac{1}{s-1}} \ge \liminf_{x\to\infty} \frac{S(x)}{x/\log{x}}.$$ This implies the assertion in question. The copy of the Handbuch I am looking at carries a date of 1909.
OLD VERSION:
Hasse mentions in the first (1950) edition of his Vorlesungen uber Zahlentheorie that if the natural density of a set of primes exists, so does the Dirichlet density, and they are equal. He calls the proof "verhältnismäßig einfach" (relatively easy), saying it is a generalization of Abel's continuity theorem. So he clearly has in mind the partial summation proof Lucia alluded to.
More interestingly, Hasse claims the converse is true (Dirichlet density exists ==> natural density exists); this is corrected in the 2nd (1964) edition. The error is discussed in Paul Bateman's MathSciNet review.
This statement is proved in detail in Tenenbaum's book "Introduction to analytic and probabilistic number theory". See Theorem 2 in Section III.1.2. See also Theorem 3 in Section III.1.3, where it is proved that the analytic density is the same as the logarithmic density.
Personally, I find the mentioned Theorem 2 rather simple, while Theorem 3 somewhat subtle. Also, I don't know who made these observations first.
Added. Actually, Theorem 2 compares the logarithmic density with the natural density, while Theorem 3 compares the logarithmic density with the analytic density. Also, these are not relative densities but true densities, but I am sure a generalization would be straightforward.
Scholz (in Jahresbericht der Deutschen Mathematiker-Vereinigung 45, 1935, p. 110, (Aufgabe 207) https://www.digizeitschriften.de/dms/toc/?PID=PPN37721857X_0045
https://www.digizeitschriften.de/download/PPN37721857X_0045/PPN37721857X_0045___log39.pdf
poses the problem below.
Given a set $M$ of positive integers $m_1<m_2< m_3 <...$ with natural density $\lim_{n \rightarrow \infty} \frac{n}{m_n}=h$.
Take the set $P=\{ p,...\}$ of those integers (e.g. in base 10) with $m_1, m_2, ...$ digits, which obviously have no natural density. Prove that it has a Dirichlet density $\lim_{s \rightarrow 1} (s-1) \sum_P p^{-s}=h$. ...
This shows that Scholz was certainly aware of the distinction between natural and Dirichlet density, by oscillating examples, similar to those alluded to (by Salvo) in the more modern literature.
For completeness, the Solution by Stöhr is in the 1938 volume https://www.digizeitschriften.de/dms/toc/?PID=PPN37721857X_0048
https://www.digizeitschriften.de/download/PPN37721857X_0048/PPN37721857X_0048___log26.pdf