Semi-Euclidean plane means that the sum of angles of triangle is two right anles,which correspnds to the Euclidean parallel property. While semi-elliptic plane means the sum is larger than two right angles, however, it doesn't correponds to elliptic parallel property(every two lines intersect) since the axiom of order and congruence ensures that we can draw at least a line parallel to a given line, so I am considering what parallel property it holds?
It doesn't seem correct to say that the semi-Euclidean property is corresponds to the Euclidean parallel property because the first is not sufficient for the second in Hilbert planes.
See Greenberg's paper again, pg 207:
C. Given any segment PQ, line l through Q perpendicular to PQ, and ray r of l with vertex Q, if θ is any acute angle, then there exists a point R on r such that $\angle PRQ < θ$.
Theorem 1. A Hilbert plane is Pythagorean if and only if it is semi-Euclidean and statement C holds.
(A Hilbert plane is just one satisfying the first 13 plane axioms, but not necessarily the parallel axiom.)
So you can see, there exist semi-Euclidean Hilbert planes without that axiom of parallels.
Finally, there exist semi-elliptic Hilbert planes which satisfy the axioms of incidence, order and congruence, and such a plane cannot also have the elliptic parallel axiom as you noted. So there is not a connection like the one you are describing.