The fact that it is in Article I of the Constitution is a clue, and in context, the prescription is clearly directed to Congress (as opposed to Section 10 which is clearly directed to the States). And, the fact that Congress passed a measure enabling Lincoln's action is another clue. The Republic is suspended when and where the president successfully asserts the unilateral power to suspend habeas corpus.
It ruled that denial of habeas corpus was illegal in areas where the federal courts operated openly. It did not say that habeas corpus could not be suspended.
My post said basically the same - that Milligan's (but not all) denial of habeas petitions was unconstitutional.
The fact that Article I contains restrictions on the powers granted the states shows that it is not directed at Congress alone. Unlike other clauses, which specifically state that Congress alone has the power to do something, Article I, Section 9, Clause 2 does not state who may suspend habeas corpus. A case may be made that the founders meant this power to be exercised only by Congress but until the Supreme Court weighs in on the matter the issue of just who may suspend it remains unsettled.