Coleman-Gross height pairings and the $p$-adic sigma function

We give a direct proof that the Mazur-Tate and Coleman-Gross heights on
elliptic curves coincide. The main ingredient is to extend the Coleman-Gross
height to the case of divisors with non-disjoint support and, doing some
$p$-adic analysis, show that, in particular, its component above $p$ gives, in
the special case of an ordinary elliptic curve, the $p$-adic sigma function.
We use this result to give a short proof of a theorem of Kim characterizing
integral points on elliptic curves in some cases under weaker assumptions. As a
further application, we give new formulas to compute double Coleman integrals
from tangential basepoints.