Elementary number theory is better defined by its focus of interest than by its methods of proof. For this reason, I rather like to think of it as classical number theory. It deals with integers, rationals, congruences and Diophantine equations within a framework recognizable to eighteenth-century number theorists. Algebraic number theory does not qualify because of its level of abstraction, even though algebraic numbers were sometimes applied to particular problems in number theory before the nineteenth century. Analytic number theory is not only distinguished by the use of complex and harmonic analysis (for many problems these are by no means indispensable), but even more by the modern emphasis on counting the number of solutions to number theoretical problems approximately. In the eighteenth century they also liked to count the number of solutions when they could, but they wanted exact answers, which severely limited the range of counting problems that they could solve. It is true that Dirichlet brought analysis into number theory, but he also counted the number of divisors of integers approximately by averaging, and Gauss before him had done the same for class numbers and genera of binary quadratic forms, as one can see from remarks in article 301 in the Disquisitiones Arithmetica (but he never published his proofs). And Legendre in 1808 published an approximation to the counting function \pi(x) of the primes, which was the start of the line of development that led to the Prime Number Theorem (Gauss had also found an approximation, but this was published only in 1863 in his collected works). The systematic acceptance of approximate answers in number theory really is a nineteenth century development. It is obvious from his interests and techniques that Euler could have found many such results if he had wanted. In 1838 when Dirichlet wrote his first paper on the approximate average number of divisors, the cupboard was so bare that he could only cite the remarks of Gauss in article 301 and Stirling's and allied formulas (!) as prior work to motivate his own.
There is a field that was absolutely central to number theory from its earliest days but for which elementary tools no longer suffice by themselves - Diophantine analysis. Actually, this transition of Diophantine analysis from elementary to non-elementary status began in the nineteenth century with the use of algebraic number theory, and gathered force in the twentieth century. But of course, there is also a huge amount of Diophantine analysis by classical techniques from the nineteenth and twentieth centuries.