How to define a 'trait' objectively?
I think you this paper is what you need. The many concepts of trait are discussed in the functional ecology background. But basically, a "trait" can be defined as 


  " 'Functional traits’ are defined as morpho-physiophenological
  traits which impact fitness indirectly via their effects on growth, reproduction and survival, the three components of individual performance." (Violle et al. 2007)


Which are not so different from the definition you posted. 
Some organisms have an improved methodology to collect and measure traits, like plants. But this is not common to all organisms. 
As for the method to correct for phylogenetic correlation, this is a whole different field called Phylogenetic Comparative Methods. To choose the method you need to know the kind of data you have and the question being asked.