Why does a word like “homemade” not constitute false advertising?
It is not amount to false advertising when its mention in the advertisement matches the common, ordinary usage of the term. In that case, the advertisement does not deceive or mislead the customers. The law only sanctions practices that are deceptive, not those which are substantially truthful.
From another standpoint, see Black's Law Dictionary entry for Home Office:
As applied to a corporation, its principal office within the state or
country where it was incorporated or formed.
This legal definition reflects that the word home does not necessarily require or imply personal residence.
If homemade doesn't imply made at home then what's the difference
between something that's homemade and something that isn't?
An item which is supplied (whether substantially or in its entirety) by an external entity would not be considered homemade.
In the context of restaurants, it is obvious that the ingredients are supplied from outside and therefore are not homemade (or homegrown). But if most or all of the preparation of a dish takes place at the facility, it would reasonably meet the criterion of being homemade. This is in contrast with a scenario where the labor of preparation is minimal because the product is delivered to the facility practically in its final, consumption-ready state.