To repeat, I think you are getting confused between the modes of (free) vibration of an object, and the components of a Fourier series for a periodic motion.I was really talking about the definition of fundamental frequency itself. I can only see it being a useful concept if it is defined so that all frequencies produced by the instrument are truly integer multiples of the fundamental frequency. I can't think of any other useful definition.
If you personally want to define "fundamental frequency" your way, that's fine - so long as ypu realize that by your definition, a guitar and a piano are NOT "musical instruments", to give just two examples.
If you disagree about the physics, sorry, but what you learned about vibrating strings in a high school level course on sound isn't the whole truth.
See http://www.simonhendry.co.uk/wp/wp-content/uploads/2012/08/inharmonicity.pdf for some real world data, on pianos.