When Is the Golf Ball Considered on the Green?

[Surgehigh]

Our normal foursome tracks putts. Today I had a ball resting on the fringe but the ball clearly extended over the the line green/fringe and on the green. I claimed I was not on the green, but was told if any part of the ball extends on the green.... I am on the green and putting. My playing partner even pulled up the rules book and read the definition of a ball on the green. It didn’t make sense to me. I need help understanding....when is a ball considered to be on the green?  A side note is we don’t play for money, but bragging rights are very important in the club after our round. 

[Missouri Swede]

Rule 13.1a 

Quote

When Ball Is on Putting Green
A ball is on the putting green when any part of the ball:
    •    Touches the putting green, or
    •    Lies on or in anything (such as a loose impediment or an obstruction) and is inside the edge of the putting green.
If part of the ball is both on the putting green and in another area of the course, see Rule 2.2c.

Which part doesn’t make sense?

[Rulesman]

any part of the ball touches the putting green

That means actually touching the grass surface

not

clear of the surface but overlapping the edge.