In set theory, a tree is usually defined as a partial order such that the set of elements below any given one is well-ordered. I am interested in the class of partial orders $P$ such that for every $p \in P$, the set of $q \leq p$ is just linearly ordered. Does this have a name?
They are also called trees.
In that terminology, trees of your first kind are known as the well-founded trees, since they are trees where the tree order is well-founded (and well-founded linear orders are the same as well-orders).
I think that the situation is that because set theorists are mainly interested in the well-founded case, the terminology evolved to drop the adjective from well-founded trees.
There are many competing definitions of tree in mathematics, not all equivalent. For graph-theorists, for example, a tree is a certain kind of cycle-free graph.
Upgraded from a comment:
After a little bit of searching, the notion of prefix order seems to be relevant; if for no other reason than that it appears to fit the required definitional niche.
(Also, it seemed worth pointing out the notion of prefix-order is precisely that of a "first-order tree".)