I agree with Matt F., there are example of metrics such that their geodesics are the same (as parameterized curves) but the metrics are different; in particular one can not use the parameterized geodesics to recover the metric in the general case.

I ask you to comment on the following two questions: Does your definition of the distance satisfy the triangle inequality? Do the curves $\gamma$ are distance-minimising, and if yes, why? Thank you

The second way, i.e., the definition of the Brownian motion as the limit of a sequence of random walks requires the following data: at every point you should have a measure on the tangent space corresponding to the choice of random direction. This field of measures is sometimes called "distribution of random directions". If two Riemannian Brownian motions are constructed by the same distribution of random directions, then their generators have the same symbols so difference between generators is a drift.

How do you define a Brownian motion corresponding to a metric? A possible way is to define it by a formula which includes the riemannian metric as an ingredient. In this case you need to compare the outputs of these formulas for two different metrics, and see whether the difference is a drift. One can define it as the limit of the sequence of random walks. A reference is Jørgensen, E. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 32(1–2), 1–64 (1975). A recent reference is my (open access) paper link.springer.com/article/10.100/s12220-021-00723-z .