I am looking to understand when a comma, involving a prepositional phrase, shifts from being a mere stylistic choice to a factor that determines the strict grammaticality of a sentence. For standard descriptive endings, a comma feels purely optional or stylistic:

• "We decided to meet at the park, near the old oak tree."
• "We decided to meet at the park near the old oak tree."

However, when we move the phrase to the front or introduce structural ambiguity, the lack of a comma seems to create structures that feel syntactically invalid or fundamentally broken. For example:

• Case 1 (Garden-path parsing): "While eating, the cat scratched at the door." vs. *"While eating the cat scratched at the door."
• Case 2 (Inversion/Subject roles): "Inside, the house was warm." vs. "Inside the house was warm." (where "Inside the house" might be misparsed as the subject of "was").

Is there a formal grammatical rule in English syntax where the absence of a comma before or after a prepositional phrase moves a sentence from "stylistically awkward" to strictly ungrammatical? Or does punctuation exist entirely outside the bounds of syntactic grammaticality?