Using PARI, I recently extended Taylor's computations based on Elkies' parameterizations and found four examples in quadratic fields of non-torsion points whose heights are smaller than those on Taylor's website.

As in Elkies' parameterization, $P=(0,0)$ for each example.

a, c and u are also as in his parameterization.

${\mathbb K}={\mathbb Q}(\sqrt{d})$ and
hgtRatio=2hgt(P)/$\log \left( {\mathcal N}_{{\mathbb K}/{\mathbb Q}} \left( \Delta_{E} \right) \right)$.

w=quadgen(d), if d is congruent to 1 mod 4

and w=quadgen(4*d) otherwise.

hgt=0.000681, hgtRatio=0.00001794, d=10, a=3 - w, u=1 + 1/2*w, c=8/9,

a1=-118 - 53*w, a2=2520 + 1044*w, a3=-755640 - 239040*w

hgt=0.001466, hgtRatio=0.00003851, d=7, a=2 + w, u=1/7*w, c=343,

a1=-16422 - 6207*w, a2=43423359 + 16412487*w, a3=-1276637821704 - 482523741504*w

hgt=0.001481, hgtRatio=0.00003735, d=33, a=1, u=-3 + w, c=1/3,

a1=-123 + 35*w, a2=-471018 + 139674*w, a3=435016512 - 128997696*w

hgt=0.001597, hgtRatio=0.00005050, d=7, a=1/3 + 1/3*w, u=-1/2, c=288,

a1=443 + 167*w, a2=-178080 - 67308*w, a3=6681612 + 2525412*w

The example above with the smallest height also has the smallest hgtRatio, 0.00001794.

The second smallest value of hgtRatio occurs for the following example:

hgt=0.001822, hgtRatio=0.00002038, d=7, a=1/7*w, u=-5/6 + 1/6*w, c=13608,
a1=4081 + 1423*w, a2=-33595908 - 12371892*w, a3=63747980088 + 23913434976*w