The Laplace transform turns linear differential equations into algebraic ones. Multiplication by $s$ is the operation corresponding to differentiation wrt to $t$ in the other domain. Maybe you should think of it as an operator, not a quantity like a generalized frequency.
In an AC circuit, there's a power source with sinusoidal voltage or current, and other elements that are proportional to it, its derivative, or its integral. Resistors, inductors, capacitors. In this case, $s=jw$ because the source is sinusoidal by design, and the derivative of $e^{jwt}$ is $jw e^{jwt}$. In other applications, we can't assume sinusoids everywhere, and $s$ has nothing to do with frequencies.
You might like Wilbur LePage's book, published by Dover.
So why these two different terms?
Because the history of mathematical terms is long and complicated. At least we stopped talking about monogenic functions and regular functions, which are two more terms for the same concept (as far as complex analysis is concerned). Quoting HOMT site:
In modern analysis the term ANALYTIC FUNCTION is used in two ways: (of a complex function) having a complex derivative at every point of its domain, and in consequence possessing derivatives of all orders and agreeing with its Taylor series locally; (of a real function) possessing derivatives of all orders and agreeing with its Taylor series locally.
Since the first usage is so popular (due to the ubiquity of power series in complex analysis, where they exist for every differentiable function), one will often say real-analytic when referring to the usage of the second kind.
Also from HOMT, an explanation of what analytic meant in the less rigorous age of analysis:
[In Lagrange's] Théorie des Fonctions Analytiques (1797) [...] an analytic function simply signified a function of the kind treated in analysis. The connection between the usage of Lagrange and modern usage is explained by Judith V. Grabiner in her The Origins of Cauchy’s Rigorous Calculus: "For Lagrange, all the applications of calculus ... rested on those properties of functions which could be learned by studying their Taylor series developments ... Weierstrass later exploited this idea in his theory of functions of a complex variable, retaining Lagrange’s term "analytic function" to designate, for Weierstrass, a function of a complex variable with a convergent Taylor series."
As for "holomorphic": in complex analysis we often encounter both Taylor series and Laurent series. For the latter, it matters very much whether the number of negative powers is finite or infinite. To enunciate these distinctions, the words holomorphic and meromorphic were introduced. Meromorphic allows poles (i.e., finitely many negative powers in the Laurent series), while holomorphic does not. From a certain viewpoint (the Riemann sphere), meromorphic functions are no worse than holomorphic ones; while at other times, the presence of poles changes the situation.
Best Answer
Yes, these are slightly different uses of the word (just like how in any language the same word can have multiple meanings based on the context). The domain of a function is just a set. Apriori, no further restrictions are needed. The domain of a function can even be the following set of "symbols" $E=\{\ddot{\smile},\ddot{\frown}, \ddot{\smile}\ddot{\smile}, @, \#\}$. THe domain of a function does not have to be a subset of complex numbers or real numbers or anything.
The use of "domain" to mean a connected open subset of $\Bbb{C}$ is indeed a different meaning to above. I've also heard people use the term "region" to describe a connected open set. For example, people use this terminology when referring to things like "domain of integration/region of integration" etc.
Sometimes, of course, if you're considering a function $f:U\to\Bbb{C}$, where $U\subset \Bbb{C}$ is open and connected, then we can say "the domain of $f$ is a domain", where of course the two uses of the word "domain" in this same sentence have different meanings. Based on the context, you should be able to decipher which meaning is intended.