Other than in matters of spelling (which belongs to the realm of written language and is thus outside the scope of usual discussions about language and cognitio

Other than in matters of spelling (which belongs to the realm of written language and is thus outside the scope of usual discussions about language and cognition), Noah Webster did not introduce any significant differences in American English in terms of grammatical categories, syntax, etc. Any influence he might have had was in matters of diction in the sense of vocabulary and stylistic choices. If Webster's educational materials were intended to be secular, they achieved this by choosing expressions that avoid religious references, for example, not by changing the building blocks of the language to make secular thoughts easier to express. The relationship between diction and cognition may be an interesting topic in itself, but it's not what we mean when we usually discuss language and cognition, where we are concerned with the structural aspects of the language itself, e.g. its tense or pronoun system, not the stylistic choice of what we express with it.