What do we call well-founded posets whose elements have a unique height?

What do we call those well-founded posets $P$ with the property that for every $x \in P$, all maximal chains in the lowerset generated by $x$ have the same length? Examples:

• The set of all finite subsets of a (possibly infinite) set.
• The set of all finite-dimensional vector subspaces of a (possibly infinite-dimensional) vector space.
• The set of all finite-dimensional affine subspaces a (possibly infinite-dimensional) affine space.
• Any set-theoretic tree.
• Any poset that could reasonably be construed as an "abstract polytope."
• A "locally graded poset" is given here as one so that each interval $[x,y]$ is graded, which is pretty close but perhaps not equivalent to what you want. – Jair Taylor May 20 '15 at 14:46
• Actually, I think I found the right definition in another paper. See my answer below. – Jair Taylor May 20 '15 at 14:51
• "Bush, not George"? "Shrub"? "Quasi-tree"? "Well-founded poset where height is well-defined"? "Mountain"? "Gregor Clegane"? I'm starting to run out of ideas... :-) – Asaf Karagila May 20 '15 at 16:16
• @AsafKaragila, I like the first one. But I thought you did not like political jokes? For the record, I've been calling such posets "well-ranked." – goblin Jun 4 '15 at 15:17
• Joking about silly former presidents is not political. Besides, well-ranked sounds reasonable. – Asaf Karagila Jun 4 '15 at 15:37

Definition 2.9. A poset $P$ will be called locally ranked if all its principal lower ideals are ranked.