Neither source shows that President Arthur lied. They show him stating that his father "came to this country when he was 18 years of age, and resided her several years before he was married," but neither proves that he lied. In fact, both sources seem to be overall dismissive of any notion that President Arthur wasn't eligible.
“I have shaken hands with a hundred people in this city who know I was born in Vermont, at the time my father was 40 years old. My father was a native of the north of Ireland. He was what they call Scotch Irish. He came to this country at the age of 18, intending to join a maternal uncle in Cincinnati. He settled in Vermont and became a Baptist clergyman.
[....]
“You can rest perfectly assured that I was born in this country, in Fairfield, Vermont, and that I am eligible.”
The family Bible recorded the birth of Chester A. Arthur for the date of 5 October 1829. The U.S. Census records confirm the date of birth as the year of 1829. The newspaper articles describe the year of birth as on or about 1830.
The newspapers reported the death and obituary of Reverend William Arthur, D.D., the father of Chester A. Arthur as 29 October 1875.
Chester A. Arthur's Eligibility—Another Letter from Mr. A.P. Hinman—A Question of Dates
To the Editor of the Brooklyn Eagle:
If you will examine either the Albany Daily Argus or the Albany Evening Journal of Friday, October 29, 1875, you will find a notice of the death of Rev. William Arthur, D.D., aged 79 years, who was supposed to be the father of Chester A. Arthur, and I believe now that he was. [....]
Brooklyn Eagle; Date: Aug 19, 1880; Page: 4.
Subtracting the reported age of 79 years from the date of death in 1875 gives a year of birth on or about 1796.
Subtracting the 40 years of age Chester A. Arthur claimed was his father's age when he was born in the year of 1829 gives Chester A. Arthur's claim a year of birth for his father on or about 1789.
Adding the 18 years of age Chester A. Arthur claimed was his father's age when William Arthur immigrated to the United States to the year of 1789 he also claimed was his father's year of birth gives the year 1807 as the year of immigration.
Adding the 18 years of age Chester A. Arthur claimed was his father's age when William Arthur immigrated to the United States to the year of 1796 he also claimed was his father's year of birth upon death gives the year 1814 as the year of immigration.
These years are mutually exclusive, so Chester A. Arthur obviously made false claims about some of the dates. They cannot all be true dates.
Finally, the record of which documents the naturalization of William Arthur as a U.S. citizen is dated 31 August 1843. Chester A. Arthur is well documented as already being 13 years of age when his father, William Arthur, gave the oath and signed the papers to become a naturalized U.S. citizen. Consequently, Chester Arthur was born with his father's nationality as a British Subject in 1829, because the British Government did not recognize any right of a British Subject to expatriate in 1829.
Since Chester A. Arthur was an experienced legal expert when he made his statement in 1880, he was obligated to know that his birth as a British Subject made him ineligible to serve in the Office of the President as a natural born U.S. citizen, just as a natural born U.S. citizen was ineligible to serve in the British political offices. Consequently, his statement/s to the newspapers included lies to deceive readers into relying upon his fraudulent representations of being eligible to serve.