Indeed, it simply is not a phylogenetic categorization but a physio-ecological one. Tree, like shrub, liana, herbaceous, woody/non-woody are all terms solely used to place plants into functional groups based on how they grow. None of these has to do with their taxonomy.

So the question is, what is a tree and is having secondary growth necessary to be one? Because monocots, like palms are, don't have secondary growth, they use some workarounds. But why should that matter in the definition of a tree? I don't know. So yeah, a coconut palm should be considered a tree. But it hasn't got to do with phylogenetics (like explained in the article you linked).

Also, millennia ago there have been vast forests of lycopods!! Just imagine huge trees that are actually spikemosses. So why shouldn't a palm not be a tree?