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.