Mmmm.... interesting. Reading these two posts above makes me question whether we really want an Active Positive president. I think Bush would have benefited from thinking through his foreign policy decisions more carefully before dashing off gung-ho to create more problems in countries like Iraq and Afghanistan. Controversial view I know! I think Regan is more of a positive example of this kind of president - he certainly achieved massive results with his foreign policy through being active.
I would agree with the above post and add Ronald Regan to the list. He did not use his opportunities to speak to criticize the American people, but he encouraged, praised, and lifted-up the American people. Likewise, around the world, he dealt with many "talks" issues. Just after his tenure as president, the Iron Curtain completely fell. As a diplomat, he learned to work with Gorbechev (the leader in USSR at the time) as well as other countries that would have confronted us. His positive tone was firm, but welcomed other players to the table to be able to present their ideas. I remember Regan in all presidential images as a man with a smile.
In addition, Regan sought solutions for the people at home. He is credited with significantly reducing taxes for the middle class and giving small buisnesses a boost to flourish. Thus, a positive active president would look like he has a strong grasp on issues at home, is in touch with the common man at home, controls situations abroad with a firm tone but resorts only on his military in extreme cases, and maintains the image that we are moving in a positive directions with his actions and gestures.
I assume that you are talking about James David Barber's classification of presidential personality types.
If so, an active positive president would be one who tends to take action whenever possible (instead of being more cautious) and who is generally upbeat about things (as opposed to an angry man like Nixon).
Whether you like him or not, I think that George W. Bush was pretty much of an active positive person (although I don't know about the part about not taking himself seriously). Bush's actions on Iraq seem to show this.
In deciding to invade Iraq, Bush was clearly active. He went out and actively sought a chance to act. At the same time, he was relentlessly positive. He was always saying that we were winning or were about to win and that we could not possibly lose.
A positive president is one that can make decisions in a time of crisis and promote the greater good for the most people. This is an important characteristic of any president. Far too often, presidents have been awash in contemplation, and it is almost ironical with all of the advisers that a president has if he can not make a decision.