The Constitution sets no limits on "running for President", in fact, anyone may "run for President" as many times as desired.
Similarly, the Constitution is silent on "being nominated", Obama could be nominated tomorrow by the Democratic Party without involving the Constitution in the slightest.
The one thing the Constitution does not allow is for Obama to "be elected" President ("No person shall be elected to the office of the President more than twice...").
It is quite unclear who, exactly, is limited by this very poorly-drafted Amendment. Does it mean Electors, chosen under their own State's rules, MAY NOT VOTE for a certain person? Does it mean, if they do, that Congress MAY NOT COUNT those votes?
Or does it mean that, if Congress counts those electoral votes, that a person twice elected my not take the oath of office again?
It would have been simplicity itself for the drafters of Amendment XXII to write, "No person who has served two terms as President shall be eligible to the office in the future", but that's not what they wrote and it isn't what the States ratified.
If the text isn’t clear, then what is dispositive is what the ratifiers (the states) originally understood and INTENDED (which can be found in the records of the state legislatures’ deliberations leading up to the final votes).
To me, it’s pretty clear that no one can be elected as POTUS more than twice. One could reasonably infer that the intent is one cannot run for a third term. That would be a reasonable inference of the intent of the ratifers of this amendment.
However as the Left has done with the rest of the Constitution, it may choose to ignore it, change it, or convolute it as you suggest in your post. They are the experts at that. That is why I think they may try for Obama again because they think Obama’s a sure thing and Leftist morality and legality is only what advances their agenda.