how are the two definitions related? function definitions u ( x ) : this functions

I do not think the Arrow-Pratt coefficient was constructed with the differential geometry curvature notion in mind. Rather, it was coined as a way to encompass some characteristics of the utility function which have economic interpretations.
In the case of expected utility theory, economists want to identify features of the utility function which determines the level of risk aversion. Intuitively, one sees that risk-aversion depends to some extent on the "curvature" or the " degree of concavity" of the Bernoulli utility function. The question is which notion/measure of "curvature" or "degree of concavity" does the job of being a sensible measure of risk aversion.
One possible way to formalize this is to say that for a given lottery $F$, agent 1 is more risk averse than agent 2 if the amount of certain money which leaves 1 indifferent with getting the outcome of $F$ is lower than that of 2. This can be written $C_1(F)<C_2(F)$, where $C$ stands for "certainty equivalent". Now what Pratt theorem shows is that the following are equivalent:
1. $C_1(F)<C_2(F)$
2. $u_1=g\circ <!--\; \circ \; -->u_2$ for some concave function $g$
3. $r_1\ge <!--\; \ge \; -->r_2$ everywhere.
From the equivalence between 1) and 3), one sees that the the Arrow-Pratt coefficient is a sound measure of risk aversion as conceptualized in terms of certainty equivalence.
The equivalence between 1) and 2) also indicates that the differential geometry measure of curvature is not great as a measure of risk aversion. Take a function $u(x)$ with a constant curvature (in the differential geometry sense) and apply transformations $g_n(t):=t^{1/n}$ to it. At the limit, the function $g_n(u(x))$ will have zero curvature almost everywhere, although it displays infinite risk aversion.
As far as the constance of the coefficient is concerned, we have a similar story. A constant $r$ has an interpretation in risk aversion terms: the risk premium, the additional certain $\mathrm{\pi <!--\; \pi \; -->}$ that would leave an agent indifferent with lottery $[x-<!--\; -\; -->h;x+h]$ does not depend on $x$, the current wealth of the agent
Finally, the equivalence between 2) and 3) tells us that it would probably be more appropriate to speak of the Arrow-Pratt coefficient as a measure of concavity than as a measure of curvature. If one is satisfied with the idea that the concave transformation of a function $u$ makes $u$ "more concave", then we see that the "more concave" (partial) ordering is represented by the Arrow-Pratt coefficient.
All this also indicate that economists do not think about the notion of curvature in differential geometric terms.