Edit: In a comment below, Gerhard Paseman very kindly called my attention to the example of $n$-valued Post algebras, which I had not heard of prior to this discussion. Apparently these were introduced by the Polish logician Emil Post based on his studies of $n$-valued logic as an extension of ordinary 2-valued logic; the $n$-valued Post algebras are to $n$-valued logic as Boolean algebras are to 2-valued logic. There are a number of equational presentations of Post algebras; see for example this article by George Epstein from the Transactions of the AMS.
Perhaps I can use this very example to follow Gerhard's advice, and deliver an attempted sales pitch specifically to him. :-) I hope it's not considered off the topic; it is meant to illustrate the basic conceptual simplicity of the Lawvere-theory way of thinking.
If my suspicions are right, the Lawvere theory of $n$-valued Post algebras is nothing but $Fin_{n^\bullet}$, the category of finite sets whose cardinality is a power of $n$. Or, in other words, that such Post algebras can be identified with functors
$$Fin_{n^\bullet} \to Set$$
that preserve finite products. Certainly for anyone used to Lawvere theories, this gives a very tidy description, and this description makes it easy to see the essential equivalence between $n$-valued Post algebras and Boolean algebras, as in the Karoubi envelope analysis above (which uses nothing more complicated than idempotent functions between finite sets).
For those not used to Lawvere theories (and of course this is assuming my suspicions are correct), the more traditional syntactic descriptions given by Post and his followers can be extracted from this categorical description. The rough idea is this. If $X$ is a Boolean algebra, and if $P(n)$ is the power set of a set with $n$ elements, then the elements of the corresponding $n$-valued Post algebra are simply Boolean algebra homomorphisms $P(n) \to X$. Let $X(n)$ denote this set. The $m$-fold cartesian product $X(n)^m$ is naturally identified with $X(n^m)$, the set of Boolean algebra homomorphisms $P(n^m) \to X$. Then, we may describe the clone of Post algebra operations: the $m$-ary operations $X(n)^m \to X(n)$ in the clone are in one-to-one correspondence with functions $f: [n^m] \to [n]$ from an $n^m$-element set to an $n$-element set. Namely, each function $f: [n^m] \to [n]$ induces a Boolean algebra map $f^{-1}: P(n) \to P(n^m)$ (the inverse image $f^{-1}$ takes a subset $S \subseteq [n]$ to $f^{-1}(S) \subseteq [n^m]$); then the corresponding Post algebra operation $X(n^m) \to X(n)$ sends a Boolean algebra map $\phi: P(n^m) \to X$ to the composite
$$P(n) \stackrel{f^{-1}}{\to} P(n^m) \stackrel{\phi}{\to} X.$$
This gives the clone; giving an explicit description of a set of operations and identities for the theory in a universal algebra sense is pretty much the same thing as giving a combinatorial analysis of functions between sets of cardinality a power of $n$, in other words how to generate such functions from a smaller class using cartesian products and composition.
I think with that clue, we can figure out what is going on in Epstein's paper. The constants of the theory are given by functions $[1] \to [n]$, so there are $n$ of them. These are the $e_i$ of his paper. Next, unary operations of the clone are given by functions $[n] \to [n]$; for his purposes, Epstein selects $n$ of them, and calls them $C_i$ ($i = 0, \ldots, n-1$). From our point of view, they are uniquely specified by how they act on the constants, since a function $[n] \to [n]$ is uniquely determined by how it acts on functions $[1] \to [n]$. Epstein's $C_i$ are just the indicator or characteristic functions given by $C_i(e_j) = \delta_{ij}$ (returning the "bottom" constant $e_0 = \bot$ if $i \neq j$, and the "top" constant $e_{n-1} = \top$ if $i = j$). These together with the lattice operations $\wedge: [n]^2 = [n^2] \to [n]$ and $\vee: [n]^2 = [n^2] \to [n]$, which are again uniquely specified by how they act on constants (and defined by the expected rules if the $e_i$ are ordered by $e_i \leq e_j$ iff $i \leq j$), are shown by Epstein to generate the theory of Post algebras, but seems pretty intuitive that they generate the clone as described here by the Lawvere theory $Fin_{n^\bullet}$: it says that any finite function $[n^m] \to [n]$ can be built from a combination of meets and joins applied to the constants $e_i$ and their characteristic functions $C_j$.
Part of my point above was that all the Post algebra theories can be united under a single simple umbrella given by the category of product-preserving functors $Fin_+ \to Set$.
