Origin of Littlewood's idea about sign changes of $Li(x) - \pi(x)$ Background (skip if you like). 
Skewes and Littlewood are closely identified with the idea that $Li(x)- \pi(x)$ changes sign infinitely often, but Littlewood closed a gap in the work of Schmidt, who refined an early idea of Phragm$\acute{\text{e}}$n, who credited Chebyshev with the original idea, calling it an idea of "perhaps little importance, which nonetheless seems somewhat curious." 
Phragm$\acute{\text{e}}$n in 1891 expressed the idea by saying "there is no limit beyond which the difference $f(x)- (Li(x) -\log 2)$ does not change sign." Of the five theorems in the paper of Chebyshev he cites, the relevant one seems to be:* 

The function $\pi(x)$...satisfies...an infinite number of times the inequalities
$$\pi(x) > \int_2^x\frac{dx}{\log x}-\frac{\alpha x}{\log^n x},~~~ \pi(x) < \int_2^x\frac{dx}{\log x}+\frac{\alpha x}{\log^n x}\hspace{10mm}(1) $$ 
no matter how small the positive constant $\alpha$ or how big n. 

Chebyshev proves this theorem and uses it later to show that the best approximation of $\pi(x)$ of the form $\frac{x}{A\log x + B}$  is $\frac{x}{\log x -1},$ already a nice result. 
If we suppress almost everything but the sense of the inequalities, letting $m$ be the quantity $\frac{\alpha x}{\log^n x}$ and $d$ the difference $\pi(x) - \int_2^x\frac{dx}{\log x},$ we can re-write (1) as
$$ d > -m,~~d < m.\hspace{20mm}(2)  $$
Question:
The logic of Chebyshev in (1),(2) seems to be almost the opposite of what Phragm$\acute{\text{e}}$n was getting at. At least superficially, for infinitely many x,
Phragm$\acute{\text{e}}$n: $\hat{d} < - \hat{m}$ and $~\hat{d} > \hat{m}.$
Chebyshev:  $d > -m$ and $~d < m.$
So does Chebyshev's theorem (1) really prefigure Phragm$\acute{\text{e}}$n's idea? 
After looking through Chebyshev's paper and staring at his theorem for a while I can't convince myself that the idea of infinitely many sign changes is part of Chebyshev's theorem (or paper). 
My interest is not in deciding who gets the laurels but in finding the earliest form of the idea, which is sometimes (but not always) instructive. 
If someone can explain how/why these ideas are the same (or not) that would be great. It may be that this was not the theorem (there are five) Phragm$\acute{\text{e}}$n intended. I have looked at the others and do not see another obvious candidate.  I don't expect this question is one of general interest and provide the archive page for Vol. 1 of Chebyshev's works (page 39 is where the theorem appears) only in case anyone wants a look (no pay wall!). Thanks.
http://archive.org/details/117744684_001
*Edit: Phragm$\acute{\text{e}}$n's attribution reads: "[My result] is, as we see, a little more precise than the one we owe to M. Chebyshev [citing the paper above]." Phragm$\acute{\text{e}}$n's f(x) is Riemann's f(x) but his paper deals with fairly general differences of this sort. So I use $\hat{d}$ and $\hat{m}$ in the question because they are not identical to Chebyshev's quantities.
 A: The main goal of Chebyshev's paper is to show  that 1.08366 is not as important as was earlier thought. He refers to 1.08366 several times throughout the paper. 
If you believe in 1.08366, or more generally that $$\liminf \left(\ln  n - \frac{n}{\pi(n)}\right) >1 \tag{*}$$ then the idea of $\operatorname{Li}n-\pi(n)$ changing sign infinitely often cannot occur to you; (*) implies that $\operatorname{Li}n-\pi(n)>0$ for all large $n$. 
Chebyshev disproves (*) by showing that $$\liminf \left(\ln  n - \frac{n}{\pi(n)}\right) \le 1\le \limsup   \left(\ln  n - \frac{n}{\pi(n)}\right) $$
although he did not prove that the limit exists.
The inequalities (1) in your post can be described as follows: if we draw a (relatively) thin   neighborhood of the curve  $y=\operatorname{Li}x$ of the specified form, for any $\alpha,n>0$, then the graph of $y=\pi (x)$ will hit that neighborhood infinitely often. 
As it turns out, the graph of $y=\pi (x)$   hits the curve $y=\operatorname{Li}x$  itself infinitely often. This is something that Chebyshev neither proved nor conjectured, but one can see it as the "limit" $\alpha\to 0$ of Chebyshev's result.
Clearly, the result with $\alpha=0$ is more precise, which is what Phragmén says: "[My result] is, as we see, a little more precise than the one we owe to M. Chebyshev". He modestly put "a little more"; after all, the quote does not come from his grant proposal.
