Interesting bit of obscure history. The argument I’d use to analyze this one would go as follows:
The first state to leave the Union, South Carolina, voted for secession on December 24, 1860. The military action that is generally regarded as starting the Civil War, the bombardment of Fort Sumter in South Carolina, began on April 12, 1861. During the three-and-a-half month period separating these two events, South Carolina was in the position of having declared its independence from the Union but of not yet having started to defend that independence by force of arms. Once the shooting started, however, the issue stopped being a legalistic one and instead became a military one, with the ultimate outcome depending on whether southern arms would be able to uphold Confederate independence from the Union or whether Union arms would be able to reinstate federal government in the southern states. So if the tiny New York hamlet of Town Line did indeed vote to secede from the U.S., but took no steps to back up its claim to independence by force of arms, then Albany and Washington would have had no compelling reason to consider this action as anything other than a symbolic vote which they could simply ignore (especially since they already had their hands full with a real war in the south). It’s even possible that Albany and Washington “never got the memo” because the article is a vague about whether Twon Line ever notified the federal authorities that they had voted to secede.