This seems like an odd place to get into a serious discussion on these topics. Those opinions in the provided link are not anything I haven't seen before, I happen to be on the other side of the fence though. I will admit there isn't really a "correct" answer. Many people consider mechanics to be part of the broad topic of grammar also. In which case saying something is a part of mechanics is also saying it is a part of grammar in general. Some consider the overall topic of punctuation a part of mechanics while others do not.
The opinion represented in the provided link usually stems from an argument along the lines of this: It doesn't matter whether the first letter of a sentence is capitalized or not so far as the meaning of the sentence, Which I concede is pretty accurate. Based on that the conclusion drawn is the one that therefore capitalization is not grammatical in the sense clarity. In fact, in the spoken language we wouldn't know if a capital letter was present or not. But the silly joke I provided does clearly show in the written language how a couple capital letters can make a big difference in the connotation of the two sentences. This and other examples like it form the other side of the argument that capitals can and do provide clarity and enhance or provide meaning and thus must also be a category of grammar.
The rules for things like capitalization have changed just in my lifetime let alone over many more generations. In fact, they are changing today. I read an interesting article on that topic not too long ago.
I recently finished a book I was writing for my lodge on its history as it approaches its 200th anniversary. I compiled several past histories, the oldest was written in the 1850s. I transcribed each as is, but after doing so and reading them over a few times I decided to take liberties with the original works and updated a number of things such as capitalization, punctuation and even some basic word structure to match current conventions. I tried to stay true to the original with my edits while easing the readability for the audience. I cited them as transcribed and noted the types of alterations I made and left the reader the locations of the original works if they are interested. My point is in language things are constantly changing and as they change disagreements exist even among the experts in the field.
I prefer teaching in my field, mathematics. One might think you would not have the above issue with a topic like math but that would be far from the truth.
This is one University's English Dept.'s list of the subtopics of grammar, but another's list could, may, and most likely will vary.
Syntax and Sentence Structure
Comma Splices, Run-ons, and Fragments
Nouns and Pronouns
Commonly Confused Words
Semicolon and Colon Usage
Hyphens and Dashes
Parenthesis and Ellipses
...BACK TO THE FUNNIES NOW