In comments Aaron asked about an example of Kevin's construction.
In Kevin's comment the rational points come from the group law on the elliptic curve and $D$ is the lcm of the denominators.
An example with $10$ points will be with so large $m,n$ it will be practically unreadable on MO, so here is magma online code and example with $6$ points.
Starting with the OP $m,n=1370,2210$ got a Weierstrass model of EC.
Found $4$ generators with mwrank and worked with the first generator $P$ (could have used some of the OP points instead of finding generators).
$P$ and $2P$ gave two additional solutions to the OP and here is the result:
D= 7061463496178454923796024335945506461398736716739985
m= 68314045389769361590347449258049778039693974321579389848056766335112029004282345656156971656752753986308250
n= 110200029424372473806326907197291977713666922080795950046865294598976338758732834963581684205418676138497250
x1= 7061463496178454923796024335945506461398736716739985
x2= 162413660412104463247308559726746648612170944485019655
x3= 261274149358602832180452900429983739071753258519379445
x4= 204782441389175192790084705742419687380563364785459565
x5= 21151048023226761412428898157420887062515978056922795
x6= 35141206529629391701793045456458600088792043478723117
m:=1370;
n:=2210;
aa<x,y,z>:=AffineSpace(Rationals(),3);
C:=Curve(aa,[m-x^2-y^2,n-x^2-z^2]);
P:=C!([1,37,47]);
pc:=ProjectiveClosure(C);
E,m1:=EllipticCurve(pc,pc!(P));
m2:=Inverse(m1);
aInvariants(E);
Ep:=E!([-323231734744697104/27633477663066497041,585299700649024 /27633477663066497041]);
m2(Ep);
m2(2*Ep);