Final answer:
It is false that the Border States quickly accepted emancipation proposals and the Thirteenth Amendment; there was resistance and the Civil War needed to conclude first. Also false is the claim that Lincoln declared the war aim was to free slaves from the start; the initial aim was to preserve the Union.
Step-by-step explanation:
The statement that the Border States quickly accepted Lincoln's proposals for gradual compensated emancipation and willingly implemented the Thirteenth Amendment is false. The Border States were reluctant to adopt these measures, and there was considerable resistance to the idea of emancipation, both gradual and immediate. It took significant effort and the conclusion of the Civil War before the Thirteenth Amendment, which abolished slavery, was fully implemented.
Regarding the statement that Lincoln announced that the war was being fought to free those who were enslaved in the South at the outbreak of the war is also false. Initially, President Lincoln stated that the purpose of the Civil War was to preserve the Union, not specifically to end slavery. It wasn't until later in the war, particularly with the issuance of the Emancipation Proclamation in 1863, that freeing the slaves became an explicit war aim.