"Сказать" generally means "to say", while "говорить" is more like "to speak" or "to talk", so it's imperfective. So if you say "как по-другому говорить" it sounds weird and is not used. It actually implies a meaning (though only formally) "how to use another method for speech".A question about a couple of the corrections you made, CoffeeCup:
Why did you choose a perfective verb in "как по-иному сказать" as opposed to, maybe, govorit'?
Why did you use "Что вы думаете" as opposed to "как вы думаете"?
I just want to understand the reasons for choosing particular forms over others.
You can also use impersonal form of "говорить" - "Как правильно говорится?" - "How do you say it correctly?"
"Что вы думаете" is used when there's an object ("Что вы думаете о порядке слов в этой фразе?") and is similar to English "What do you think about the word order in this phrase?"
"Как вы думаете", on the other hand, is a complete simple sentence. "Как вы думаете, порядок слов в этой фразе правильный?" - "What do you think, is the word order correct in this phrase?"