He was just the President, replacing Montgomery who was the President and managing partner (with MIddleton assuming the latter role). I think it was always pretty clear that baseball operations was not a big part of MacPhail's portfolio, whether by design, indifference or ineptitude. He worked on the business side, he worked a lot on stadium stuff, he gave his GM a pretty free hand. He didn't hire the manager or make player moves. When someone who wasn't Klentak did that, it was the owner.
If you look at the (still not updated) team web site MacPhail was at the top of the Executive tier, while Klentak (whose name is gone) was at the top of the baseball tier. It's possible Dombrowski's name will go over the latter (and then when MacPhail's contract expires Dave Buck will become President). Like the Dodgers, who have Stan Kasten as president and Friedman as baseball ops and then a GM (though nobody has that title now).
Given Dombrowski's history, he is likely to be the exact opposite of MacPhail, he'll be 90% hands-on baseball and his "GM," if they even get someone with that title, will be a glorified assistant. Which could also just be Rice for a year, of course.