Not true. In early December Anderson had been visited by Major Don Carlos Buell who gave him written instructions to act in the manner he thought best for the safety of his command. That included moving his men to a single facility.
Buchanan slumped into a chair. "My God!" he cried wearily. "Are calamities ... never to come singly! I call God to witness -- you gentlemen better than anybody else know that this is not only without but against my orders. It is against my policy." ...
Buchanan gives a somewhat different view on things in his letter to the South Carolina commissioners. Link
If Anderson had stayed in Fort Moultrie, the country (countries, actually) might have been spared a war.
Doubtful. If Anderson had remained in Moultrie then his position would have been attacked by the South Carolina mob and militia, as he was warned would happen. Moultrie at the time was indefensible, with civilian buildings being built up to the walls and overlooking the parapet.
After hearing about the Buell instructions after they were sent, Buchanan was "staggered" according to Klein. He objected to the "defend yourselves to the last extremity" of the Buell letter. He dictated a response to Anderson saying "it is neither expected nor desired that you should expose yourself or that of your men in a hopeless conflict in defense of the forts." Buchanan's instructions were sent.
Governor Pickens had demanded in a letter to Buchanan that state troops be placed in Fort Sumter. This was before Anderson moved to Sumter. Buchanan offered the 'status quo in the forts' agreement to forestall Pickens. Anderson later violated that by moving to Sumter.