Lincoln's final orders to both Fox and all ship captains, dated April 6 and issued through Secretaries Cameron & Wells, said the same thing Lincoln told SC Governor Pickens: resupply-only, if not resisted.
You can read them yourself at the link above.
“SIR: This letter will be handed to you by Capt G.V. FOX, ex-officer of the navy, a gentleman of high standing as well as extraordinary nautical ability. He is charged by high authority here with the command of an expedition (under cover of certain ships of war) whose object is to reinforce Fort Sumter.”
Your own source puts the lie to your assertion.