Early history of lower bounds on the prime counting function Let $\pi (x)$ be the number of prime numbers less than or equal to $x$. 
Euclid's proof of the infinitude of primes gives a horrible lower bound of the type $ \pi (x)>> \sqrt{\log{x}} $. Somewhat surprisingly, Euler's proof of the divergence of the sum of reciprocals of primes only gives $ \pi (x)>> \log{x}$ (see problem 8 of http://www.math.uiuc.edu/~hildebr/531.fall05/hw3.pdf). 
Was anybody able to get a lower bound better than $\log{x}$ before Chebyshev got the order of magnitude right in ~ 1850?
 A: Incredibly, Gauss in 1793 conjectured $\pi_k(x)\sim \frac{x (\log\log x)^{k-1}}{(k-1)!\log x }.$ Gauss, Werke vol. 10 (Gottengen (1917),11.* In his Handbuch Landau discusses Gauss in one of the earlier sections. If I have the German right he ascribes to Gauss that there is a function A(x) with 
$\pi(x) \sim \frac{x}{\log x - A(x)}$ and the conjecture that $\lim_{x \to \infty} A(x) = 1.$ 
Landau adds, "Aber beweisen hat Gauss nichts..." So it was maybe conjecture based on extensive calculation. 
Gauss does count, as I doubt he would have speculated at all if there were already a proof extant. Given his prominence, I expect he would have been aware of one. 
Landau also quotes Legendre in 1808 to the effect that "with a very satisfying precision" $\pi(x) \sim \frac{x}{\log x - 1.08366}.$ 
The OP asks about lower bounds and I think Landau gives a pretty complete review of the situation as it existed prior to 1850. Apart from what is mentioned in the OP I don't find anything. Doesn't mean it's not there...
Edit: Chebyshev did give a proof (possibly conditional) that $$\lim \sup_{x\to \infty} \pi(x) \geq \frac{x}{\log x}. $$ This is also from Landau and I am fairly confident that there is nothing much earlier.
*This assertion is from the first page of E.M. Wright, (1954), A Simple Proof of a Theorem of Landau,  Proc. of the Edinburgh Math. Soc., 9, pp. 87-90. 
