Well, it seems that all problems were related to my inattention. It is not in the size of the matrix, but in the type of the matrix - it should be symmetric. I was inaccurate to make a mistake in just one value in my matrix. If one uses corrected input matrix, which looks like this:
M = {{0,1,1,1,1,0,0,0,0,0,0,0,0,0},
{1,0,0,0,0,1,1,0,0,0,0,0,0,0},
{1,0,0,0,0,0,1,1,0,0,0,0,0,0},
{1,0,0,0,0,0,0,1,1,0,0,0,0,0},
{1,0,0,0,0,1,0,0,1,0,0,0,0,0},
{0,1,0,0,1,0,0,0,0,1,1,0,0,0},
{0,1,1,0,0,0,0,0,0,0,1,1,0,0},
{0,0,1,1,0,0,0,0,0,0,0,1,1,0},
{0,0,0,1,1,0,0,0,0,1,0,0,1,0},
{0,0,0,0,0,1,0,0,1,0,0,0,0,1},
{0,0,0,0,0,1,1,0,0,0,0,0,0,1},
{0,0,0,0,0,0,1,1,0,0,0,0,0,1},
{0,0,0,0,0,0,0,1,1,0,0,0,0,1},
{0,0,0,0,0,0,0,0,0,1,1,1,1,0}}
or like this:
bb = {{2, 3, 4, 5},
{1, 6, 7 },
{ 1, 7, 8 },
{ 1, 8, 9 },
{ 1, 9, 6 },
{ 2, 5,10,11 },
{ 2 ,3 ,11 ,12 },
{ 3 ,4, 12, 13 },
{ 4 ,5 ,13 ,10 },
{ 6 ,9, 14 },
{ 6 ,7 ,14 },
{ 7 ,8, 14 },
{ 8 ,9 ,14 },
{ 10, 11, 12,13}}
the same script by @Szabolcs Horvát works perfectly! And this totally solves my problem! Once again, thank you, Neil - you actually helped to solve the title problem of the post. Szabolcs, thank you for the script for finding face vertices from the face adjacency graph, which I found in the community, and for the beta version of IGraph/M - I will contact you in case I have any feedback information.
Best regards, Anton