A relative monad is defined in Altenkirch et al. essentially as a pair of functors $J,F:\mathcal{C}\to\mathcal{D}$, a natural transformation $\eta:J\to F$, and an operation $(-)^{*}:\mathcal{D}(J-,F-)\to\mathcal{D}(F-,F-)$ with conditions that give a composition rule $\mathcal{D}(JB,FC)\times\mathcal{D}(JA,FB)\to\mathcal{D}(JA,FC)$ in which the components of $\eta$ are the identities. (I'm paraphrasing a bit; the paper doesn't start with the assumption that $F$ is functorial or $\eta,(-)^{*}$ natural, but derives these as theorems.) That is, the correct notion of the Kleisli category of a relative monad seems to be what the definition of a relative monad was tailor-made to give.
An Eilenberg-Moore algebra on a relative monad consists of an object $X\in|\mathcal{D}|$ and an operation $\chi:\mathcal{D}(J-,X)\to\mathcal{D}(F-,X)$ such that for any $f:JA\to X$ $f=\chi(f)\circ\eta_{A}$ and for any $g:JB\to FA$ we have $\chi(\chi(f)\circ g)=\chi(f)\circ g^{*}$. As with the definition of a relative monad itself, the authors of the paper proposing this definition don't phrase what's happening in what I would think of as "categorical terms", and aren't really interested in delving into the high level consequences of this definition, so I'm wondering if there's a more conceptual (i.e. friendly to extremely abstract categorical types) way of phrasing what's happening with this version of Eilenberg-Moore algebras. I would like to set out criteria for a good answer here, but I don't know how to do this without reference to the very vague notion of what a "properly categorical" description of something is. Perhaps the following paragraph of my thinking on the topic can suggest the flavor I'm looking for.
I'm aware that the definition as given in the cited paper gives one that an algebra is at least partially a cone $\chi$ under $F\circ Q:(J\downarrow X)\to\mathcal{C}\to\mathcal{D}$ (where $Q$ is the obvious forgetful functor) such that $\chi\circ\eta Q$ is equal to the "canonical cone" under $J\circ Q$ with vertex $X$, which is likewise the case in an ordinary monad where $J=id_{\mathcal{C}}$. The $\chi(\chi(f)\circ g)=\chi(f)\circ g^{*}$ condition must say something more than this, though, as I see no way of pulling this back out from only the description in terms of cones just given. If relative monads always arose from a (lax) monoidal structure on $\mathcal{D^C}$ I would look for the extra structure there, but the conditions for this given in Altenkirch et al. don't hold generally enough (and in particular don't hold in the cases I'm studying).