Let's see if I understand this -- general & future president Andrew Jackson first suspended habeas corpus, then later, after the crisis had passed, apologized and paid a fine for it?
By contrast, Congress debated President Lincoln's actions at great length, and in the end decided to neither censure nor indemnify him for them, but to grant him Congressional authorization to suspend habeas corpus.
Seems to me, that should settle the question.
Your summary glosses over the details of what happened in the Jackson case and ignores what was different between Lincoln's and Jackson's actions.
On March 3, 1815, some two months after the Battle of New Orleans, a newspaper article appeared in the Louisiana Courier newspaper asking for the end of military tribunals and the adjudication of any charges against civilians to be returned to the regular government court system. Within a day or so, Jackson found out that a Louis Louallier had written the article. Jackson had him arrested. A petition for habeas corpus for Louallier was then sent to Judge Hall of the US District Court who issued the writ. Without Congressional approval to suspend the writ, Jackson ignored the writ and did not provide the prisoner for the habeas hearing.
Up to this point there is some similarity to the Lincoln situation. Lincoln suspended the privilege of the writ without the approval of Congress. One of his military commanders arrested someone (Merryman) for drilling a Southern-sympathizing company that was using US government weapons. Because of Lincoln's order the commander would not honor a writ of habeas corpus issued by the Chief Justice of the US Supreme Court (Taney). The military commander or his predecessor had for the same reason previously ignored a different writ of habeas corpus involving another prisoner issued by the judge of the local US District Court. And, of course, Lincoln's military later arrested newspaper writers, editors, and publishers for the content of their newspapers and held them in prison for months without charges.
On about March 5th or 6th, Jackson arrested the judge that issued the writ. Lincoln did not arrest the Chief Justice, but there are several references in the literature that say an arrest order for the Chief Justice was indeed prepared in the Lincoln case but never executed. However, a Maryland judge was arrested in his courtroom for following Taney's legal reasoning and DC judge was held in house arrest so that he could not rule on another habeas case.
On March 11, Jackson sent Judge Hall out of New Orleans and released him. Two days later he released Louallier. On March 31, Jackson appeared in court to answer charges that he had arrested Judge Hall. Jackson did not respond to the charges. The court then issued the judgment that Jackson pay a $1,000 fine. Jackson went back to his quarters and wrote a check for the amount. As I posted above, Jackson publicly said:
Considering obedience to the laws, even when we think them unjustly applied, as the first duty of the citizen, I did not hesitate to comply with the sentence you have heard pronounced, and I entreat you to remember the example I have given you of respectful submission to the administration of justice.
In contrast, Lincoln did not submit to the administration of justice and the Constitution like Jackson had. He did not follow Jackson's precedent, and consequently people were held for more than a year in prison without charges in areas where the regular government courts were functioning.
I do not believe that Lincoln ever cited the Habeas Corpus Act of 1863 in any of his subsequent pronouncements of the suspension of the writ. Apparently, he still acted as though he had complete authority to do what he wanted. He had assumed the duties of Congress regarding habeas corpus and other Congressional responsibilities. He assumed the duties of the judicial branch of government by ignoring a legal order issued by the Chief Justice. He tossed people in jail to be tried by military courts in areas where the regular government courts were functioning.
It was Madison who said the following in Federalist 47:
The accumulation of all powers, legislative, executive, and judiciary, in the same hands, whether of one, a few, or many, and whether hereditary, self appointed, or elective, may justly be pronounced the very definition of tyranny.
That would seem to fit Lincoln's actions. I am also reminded of the ruling of the Supreme Court in Ex Parte Milligan (1866). Significantly, it was Lincoln’s friend and executor of Lincoln's estate, Supreme Court Justice Davis, who issued the unanimous ruling against Lincoln's policies which said in part:
The Constitution of the United States is a law for rulers and people, equally in war and in peace, and covers with the shield of its protection all classes of men, at all times and under all circumstances. No doctrine involving more pernicious consequences was ever invented by the wit of man than that any of its provisions can be suspended during any of the great exigencies of government.
Here are a couple of references about the Jackson case: Despotism and Battle of New Orleans