Ah, leeneia, 'rules', 'law' and 'grammar' - the source of so much fruitless argument.

Gnu, dear, these people actually wrote, "Find the perfect word every time with context-optimized synonym suggestions".

This is a single sentence, therefore the period belongs at the end, outside the quotation mark. Contrast that with your final paragraph:

(Yes, I know the law says the period belongs inside the quotation mark. I reject that as illogical.)

Here, the complete sentence is in parenthesis, so that is where the period belongs.

Your post is a beautiful example of the variation in placement of a period.

Phil