There's a major difference. Bush was trying to find and kill OBL while Bammy let al Baghdadi loose.
I thought the subject of this OP was concerning our standard protocol norms for notifying an ex-president however let's not let facts get in the way.
In my view as well as many others, Bush actually outsourced to the Afghan Army at Tora Bora which allowed OBL safe passage into Pakistan. It is well known that Afghan Commander Ghamsharik and some of his fellow Khugani tribesmen offered the Arabs "safe passage" out of Tora Bora. We can argue if this was the correct decision at the time however the fact remains that we had OBL surrounded.
As far as al Baghdadi, per the Pentagon, he was released in 2004:
“Ibrahim Awad Ibrahim Al Badry, also known as ‘Abu Bakr al-Baghdadi’ was held as a ‘civilian internee’ by U.S. Forces-Iraq from early February 2004 until early December 2004, when he was released. He was held at Camp Bucca. A Combined Review and Release Board recommended ‘unconditional release’ of this detainee and he was released from U.S. custody shortly thereafter. We have no record of him being held at any other time.â€
and
al-Baghdadi’s detainee records through a 2015 Freedom of Information Act request. The official documents show that al-Baghdadi was captured on Feb. 4, 2004, and that the date of his release was Dec. 8, 2004.
https://www.businessinsider.com/abu-bakr-al-baghdadi-declassified-iraq-prison-file-2015-2