What a position means is far from standardized in the industry. In general, game producers in big publishers in the US have little to no impact on the actual product. They're facilitators, keeping different departments abreast of things and making sure that all the pieces come together. And again, this is in general; this will vary between publishers and developers. And Japan probably has a different structure than the US in terms of the role of the producer.
One constant however is the rule of Executive Producer. Just as with movies, EPs often have little actual power or role in the production. Sometimes, they're purely ceremonial, used as a way to credit someone for part of the work. Some EPs do have some authority, but this is primarily through their role as facilitating communication between the upper level executives and the people actually making the work. When people speak of "Executive Meddling", the EP is often the go-between or facilitator for this.
Thus they can gain de-facto power by simply lying or misrepresenting how things are going in production to the people with actual power. Some EPs have veto power, some don't. Even so, the EP's power is usually via the commandment: "Do this!" Exactly how it gets done tends to be up to the developers. Though again, there are exceptions.
Basically, you can say nothing about how much influence over the production of those games that Sakaguchi had. At best, you can vaguely say that he probably had more direct, intimate control when he was Director, some control as Producer, and probably less moment-to-moment control over things as Executive Producer. Anything more than that is sheer speculation.