It is at least debatable whether "in Cases of Rebellion or Invasion the public Safety may require it," and Congress not in session, the President may constitutionally suspend habeas corpus on his own authority.
What's not debatable is that Congress does have such constitutional authority, and after lengthy debates in 1861 & 1862 finally decided in February 1863, to neither censure nor indemnify President Lincoln, but rather to permit him suspensions of habeas corpus on Congress' authority.
And that precident has been repeated several times since the Civil War.
Seems to me that should be the final constitutional word on this subject.
I think the Constitution and the founders (such as Hamilton, Madison, Jay, and others) were clear on what entity has the power to suspend the privilege of the writ of habeas corpus.
Lincoln suspended the privilege of the writ multiple times when Congress was not in session. The one possible exception to this was when Lincoln suspended it for Missouri on the same day that Congress reconvened. Perhaps he sent his proclamation out before Congress formally opened its session that day.
I thought it might be instructive to compare the actions of the two presidents with respect to their suspensions of the privilege of the writ of habeas corpus. This list does not include situations where an unauthorized person might have suspended the privilege in his local area.
Lincoln
March 4, 1861. Lincoln inaugurated as president.
April 27, 1861. Without Congressional approval, Lincoln authorized suspension of the writ near the military railroad lines in Maryland and the railroad line up to Philadelphia on April 27, 1861 . Source: Link
May 10, 1861. Without Congressional approval, Lincoln authorized the commander of US forces on the Florida coast to suspend habeas corpus on May 10, 1861. Source: Link
July 2, 1861. Without Congressional approval, Lincoln authorized the commander of US forces to suspend the writ of habeas corpus along the military line between Washington and New York as needed. Source: Link.
October 14, 1861. Without Congressional approval, Lincoln authorized General Winfield Scott and officers under his command to suspend the writ any place between Bangor, Maine, and Washington on October 14, 1861. Source: Link
December 2, 1861. Without Congressional approval, Lincoln authorized the commanding general in Missouri to suspend the writ on December 2, 1861. Congress was in session starting on December 2. Source: Link
August 8, 1862. At the direction of Lincoln and without Congressional approval, Edwin Stanton suspended the writ for draft resisters and persons arrested for disloyal practices on August 8, 1862. Source: Official Records: Series 3, vol. 2, Part 1 (Union Letters, Orders, Reports) Page 370 Correspondence, etc.
September 24, 1862.Lincoln suspended habeas corpus nationwide without Congressional approval on September 24, 1862. Source: Link
March 3, 1863. Habeas Corpus Act. Congress approves Lincoln suspending the privilege of the writ during the war.
May 30, 1863. The Governor of Indiana writes to Lincoln protesting Gen. Burnside’s Order No. 38, the order that was used to arrest Clement Vallandigham. Vallandigham requests a writ of habeas corpus. The request reaches the Supreme Court of the US, which decided it could not intervene in a military trial. The use of military trials to try civilians in areas where the regular civil courts are operating was later declared unconstitutional by the Supreme Court in Ex Parte Milligan (1866). See comment of the Indiana governor to Lincoln concerning Burnside’s Order No. 38 [Link.
September 15, 1863. Lincoln suspended habeas corpus nationwide with Congressional approval on September 15, 1863. This was ended by President Johnson on December 2, 1865. See: Link
Davis
February 8, 1861.The provisional Constitution of the Confederate States of America was adopted. It has a habeas corpus clause similar to that of the US Constitution.
February 18, 1861.Davis inaugurated as president.
March 11, 1861. The permanent Constitution of the Confederate States of America was adopted. It has a habeas corpus clause similar to that of the US Constitution.
All of 1861. Davis does NOT suspend the writ of habeas corpus although the Confederate Congress had the power to authorize Davis to suspend it under the Confederate Constitution.
February 22, 1862. Davis inaugurated as president a second time. In his inaugural address he excoriates the North for having suspended habeas corpus by executive mandate (rather than with Congressional approval). Source: Link.
February 27, 1862. The Confederate Congress authorizes Jefferson Davis to suspend the privilege of the writ for cities, towns, and military districts in danger of attack. Davis issues a series of proclamations under this act over the next couple of months (listed below). Source: Link
February 27, 1862. In a proclamation stating he has been given the authority by Congress to suspends the writ of habeas corpus, Davis suspends the writ in Norfolk and Portsmouth, Virginia. Source: Link. Source: Official Records, Series II, Volume 3, page 121.
March 1, 1862. In another proclamation Davis suspended the writ in Richmond, Virginia by the authority given him by Congress. Source: Official Records, Series II, Volume 3, page 122.
March 8, 1862. By the authority given him by Congress, Davis suspends habeas corpus in Petersburg, Virginia. Source: Official Records, Series II, Volume 3, page 123.
March 14, 1862.By the authority given him by Congress, Davis suspends the writ in five counties in Virginia. Source: Source: Official Records, Series II, Volume 3, page 124.
March 29, 1862. By the authority given him by Congress, Davis suspends the writ of habeas corpus in ten more counties in Virginia. Source: Source: Official Records, Series II, Volume 3, page 126.
April 8, 1862. By the authority given him by Congress, Davis suspends the writ of habeas corpus in the Department of East Tennessee. Source: Official Records, Series II, Volume 3, page 128.
April 19, 1862. The Confederate Congress restricts their February 27, 1862 habeas corpus act to arrests made by the authorities of the Confederate Government or for offenses against the same. The February 27, 1862 Act is set to expire October 13, 1862. Source: Link
May 1, 1862. By the authority given him by Congress, Davis suspends the writ of habeas corpus in parts of coastal South Carolina. Source: Source: Official Records, Series II, Volume 3, page 132.
May 3, 1862. By the authority given him by Congress, Davis suspends the writ of habeas corpus in five Virginia counties. Source: Official Records, Series II, Volume 3, page 132-133.
September 18, 1862. The Confederate Congress restores habeas corpus. Source: Link.
October 13, 1862. The February 27, 1862 habeas corpus act expires and the Confederate Congress renews it until February 13, 1863. Source: Yankee Leviathan by Bensel, pg 142.
February 13. 1863. The February 27. 1862 habeas corpus act expires. Source: Yankee Leviathan by Bensel, pg 142.
February 13, 1864. The Confederate Congress passes a bill allowing habeas corpus to be suspended by the President, the Secretary of War, or the Commander of Trans-Mississippi. It lasts until no more than 90 days after the next meeting of Congress. Source: Link. The act is to expire August 1, 1864. Source: Yankee Leviathan by Bensel, pg 142.