No, it does not count as an "item requested from someone else"
The question is not really clear in English. But if you take a look at the same form in French, question 1.6 is translated as such:
Articles qui vous ont été confiés par un tiers.
...which means, unmistakably this time:
Items that someone else gave you/entrusted you with
This question is designed to find out if someone asked you to take something through customs. If you are just bringing a gift for someone you can check "No", even if they asked you for that gift.
Basically, you should check "No" as long as you know what you are carrying. If you bought the gift yourself and packed it yourself, you're fine. However, if a stranger gave you "a package" to deliver to Japan, then you don't know what you are carrying, that is an issue, and in this case you should check "Yes".
The original C5360 form in Japanese seems clearer than the English version too. It says the following:
...which I would translate to:
Something a stranger gave me.
However I'm not a native speaker so I can't translate the meaning of this sentence with the same certainty as the French sentence above.