In the published version of HTT (Def 4.4.5.2) and on the nlab one finds one definition of a split homotopy coherent idempotent corepresented by a quasicategory $Idem^+_\mathrm{old}$. A few months ago, Gal Dor found an issue in Lurie's treatment of idempotents, and Lurie rewrote the section. The definition of an idempotent in a quasicategory has changed: in the updated version of HTT, it's a map from $Idem^+_\mathrm{new}$, which is simply the nerve of the free split idempotent in $\mathsf{Cat}$.
In his question, Dor suggested that these two definitions are the same, but I'm not so sure. I count 8 nondegenerate 3-simplices in $Idem^+_\mathrm{new}$, but 6 nondegenerate 3-simplices in $Idem^+_\mathrm{old}$, so I don't think they're isomorphic. I don't see an obvious bijection between the simplices of the two models.
Questions:
Are $Idem^+_\mathrm{old}$ and $Idem^+_\mathrm{new}$ equivalent quasicategories?
Is there at least a map between them?
It seems that all the foundational results about the old definition have been re-proved for the new definition (including a cohrence result -- the new Prop 4.4.5.20 which originally appeared under the old definition in Higher Algebra). Are there any results in the literature which rely on the details of the old construction?
EDIT
Just for concreteness, here are the non-degenerate simplices in dimension 3:
For $Idem^+_\mathrm{old}$:
A 3-simplex is an (unlabeled) set of disjoint nonempty subintervals of $[3]$, which is nondegenerate iff each subinterval has exactly one element and each "gap:" between subintervals (including the gaps at the beginning and end) have at most one element. So a nondegenerate 3-simplex can be specified by a set of numbers between 0 and 3 (each representing a singleton interval). They are:
$\begin{align*} \{0,1,2\}, \{0,1,3\}, \{0,2,3\}, \{1,2,3\},\{0,2\}, \{1,3\}, \{1,2\}, \{0,1,2,3\} \end{align*}$
for a total of 6 8.
For $Idem^+_\mathrm{new}$:
A 3-simplex is a sequence of 3 composable morphisms in the free split idempotent $A^{\overset{i}{\to}}_{\underset{r}{\leftarrow}} X \overset{e}{\to} X$, and it's nondegenerate if none of them are the identity. They are:
$\begin{align*} (i,r,i), (i,e,r), (i,e,e), (r,i,r), (r,i,e), (e,r,i), (e,e,r), (e,e,e) \end{align*}$
for a total of 8.
Side note: I personally find the new definition both much cleaner and much more transparent. I'm still puzzled about the motivation of the old definition.