To avoid creating new binomial posets by taking two of them with the
same factorial function (i.e., number of maximal chains in an $n$-interval)
and identifying their least elements, we
should add the extra condition that there exists a maximal chain
$\hat{0}< x _0< x _1< \cdots$ such that every element $x$ satisfies $x\leq
x_n$ for some $n$. Backelin has constructed an uncountable number of  nonisomorphic
such binomial posets, all with the same factorial function. Most of
these have the property that they contain two nonisomorphic intervals
of the same length. See 

* Jörgen Backelin, _Binomial posets with non-isomorphic intervals_
https://arxiv.org/abs/math/0508397.