For the early Church, the Lord's Day, or Kyriake, as it came to be known in Greek (and Dominicus, being exactly equivalent, in Latin), was the first day of the week, or Sunday.
Justin Martyr, First Apology, 67 (circa 155)
... And on the day called that of the Sun, all who live in cities or in the country gather together to one place, and the memoirs of the Apostles [[elsewhere called explicitly "Gospels" by Justin]] or the writings of the Prophets are read, as long as time permits; then, when the reader has ceased, he that presides verbally instructs, and exhorts to the imitation of these good things. Then we all rise together and pray, and, as we before said, when our prayer is ended, bread and wine and water are brought, and the one presiding in like manner offers prayers and thanksgivings, according as he is able, and the people assent, saying 'Amen.' Then there is a distribution to each, and a participation of that over which thanks have been given, and to those who are absent a portion is sent by the Deacons. ...
Just as we still call Sunday "Sun-Day" without believing in Sun gods, or Saturday Saturn-Day, without believing in Saturn, so did the early church, yet they had a Christian name for the day, namely, Lord's Day — Kyriake in Greek, and Dominicus in Latin. This is when the Eucharist or Mass or Divine Liturgy was held, since the first century.
Didache (first century), 14
But every Kyriake (even the [Day] of the Lord) [kata kyriake de kyriou] gather yourselves together, and break bread, and
give thanksgiving after having confessed your transgressions, that your sacrifice may be pure. But
let no one that is at variance with his fellow come together with you, until they be reconciled, that
your sacrifice may not be profaned. For this is that which was spoken by the Lord: In every place
and time offer to me a pure sacrifice; for I am a great King, says the Lord, and my name is
wonderful among the nations. (Didache, chap. 14)
The name Lord's instead of Sun's happens by way, presumably, of the word for day (hemera) being feminine, and thus the feminization of Kyriou.
So the word "day" does not need to be explicit if the word "Helios" for example, is already the name of the day, as Kyriake was for the early Church, of the same day. There is no abmiguity as to the specific day in question, for it is explicitly named. So to quibble about the lack of the word "day" is to simply be ignorant of the Greek (and Latin) way of naming days.
Even in Spanish the word for Sunday is Domingo (just as the word for Saturday is Sabbado), coming from the Latin Dominicus. Similarly, the French Dimanche, comes from the same. Likewise the Italian Domenica, Portuguese Domingo, Irish Gaelic, dé domhnaigh, Scottish Gaelic didòmhnaich, and related Latvian svētdienā (holy day), Russian воскресение (Resurrection), etc.
The Sabbath, or Saturday, was clearly distinct, for the early Church, from the new sabbath of the Lord's Day on which a new creation was wrought — they argued that just as Christ is the new Adam, and a new creation was wrought, He had a new rest from His work the day He rose from His work, on the first day of the week:
...We keep the eighth day [Sunday] with joyfulness, the day also on which Jesus rose again from the dead... (Epistle of Barnabas 15:6–8 [A.D. 74]).
... But Sunday is the day on which we all hold our common assembly, because it is the first day on which God, having wrought a change in the darkness and matter, made the world; and Jesus Christ our Savior on the same day rose from the dead... (Justin Martyr, First Apology, 67 (circa 155))**
The apostles further appointed: On the first day of the week let there be service, and the reading of the holy scriptures, and the oblation [sacrifice of the Mass], because on the first day of the week [i.e., Sunday] our Lord rose from the place of the dead, and on the first day of the week he arose upon the world, and on the first day of the week he ascended up to heaven, and on the first day of the week he will appear at last with the angels of heaven” (Didascalia, 2 (circa 225)).
The Sabbatarian has no case unless he accuses the early Church of being wrong wholesale on basic elements of worship since the beginning. But they are quite willing to be so bold.