The identity under question can be found in the paper S. Boettner, V.H. Moll The integrals in Gradshteyn and Ryzhik. Part 16: Complete elliptic integrals, pages 11-12 https://arxiv.org/abs/1005.2941
The proof given by OP is miraculously similar to the one in this paper, not only in the method used, but also in the choice of words and formatting of equations. By the way, one of the authors V. Moll, is a frequent collaborator of the OP's, and they have written numerous papers that start in the same way The integrals in Gradshteyn and Ryzhik. Part ..., e.g. https://arxiv.org/abs/1004.2440. Coincidences happen, and with some people they happen more often than with others, this is just law of probabilities. No foul play is suspected here. But it is really intriguing to know, how is the identity under question related to digamma function, as OP claims? Unfortunately his proof does not contain any digamma functions. But in the paper the identity naturally came out of some work on the elliptic integral. Please, @T.Amdeberhan, would you share your insight? It is really intriguing to know, how is digamma function related here?
Here is a screenshot of OP's own answer, just in case:
