LORD'S DAY
(he kuriake hemera):
1. Linguistic:
Formerly it was supposed that the adjective kuriakos (translated "the Lord's") was a purely Christian word, but recent discoveries have proved that it was in fairly common use in the Roman Empire before Christian influence had been felt. In secular use it signified "imperial," "belonging to the lord"--the emperor--and so its adoption by Christianity in the sense "belonging to the Lord"--to Christ--was perfectly easy. Indeed, there is reason to suppose that in the days of Domitian, when the issue had been sharply defined as "Who is Lord? Caesar or Christ?" the use of the adjective by the church was a part of the protest against Caesar-worship (see LORD; THE LORD ). And it is even possible that the full phrase, "the Lord's day," was coined as a contrast to the phrase, "the Augustean day" he sebaste hemera), a term that seems to have been used in some parts of the Empire to denote days especially dedicated in honor of Caesar-worship.
2. Post-Apostolic:
"Lord's day" in the New Testament occurs only in Revelation 1:10, but in the post-apostolic literature we have the following references: Ignatius, Ad Mag., ix.1, "No longer keeping the Sabbath but living according to the Lord's day, on which also our Light arose"; Ev. Pet., verse 35, "The Lord's day began to dawn" (compare Matthew 28:1); verse 50, "early on the Lord's day" (compare Luke 24:1); Barn 15 9, "We keep the eighth day with gladness," on which Jesus arose from the dead." I.e. Sunday, as the day of Christ's resurrection, was kept as a Christian feast and called "the Lord's day," a title fixed so definitely as to be introduced by the author of Ev. Pet. into phrases from the canonical Gospels. Its appropriateness in Revelation 1:10 is obvious, as John received his vision of the exalted Lord when all Christians had their minds directed toward His entrance into glory through the resurrection. - International Standard Bible Encyclopedia
Second Century Ante-Nicene writers used "kuriakos" for the resurrection day and it was well known to be the day when Ceasar was to be acknowledged as "Lord" and therefore it was also a test of faith for Christians whether they would acknowledge Ceasar or Christ as "Lord."
What you fail to understand that it was not called the "imperial" day but the "Lord's" day because the Emperor required the confession of faith that he was "Lord" as the god/man. He did not require a confession that he was "Emperor" as none denied that. Hence, the use of "kuriakos" reflected the confession called for on that day.