What I meant by interchangable, is that on a New Moon, there is no Sabbath
Other lunarians disagree with you, and I have read them, and spoke with them. Who is right? You or they? Lunarians as a whole are not consistent, nor are they unified as they claim, not even upon this point.
but there is still a rest.
So you say, other Lunarians say it is not merely a 'rest', but is indeed a 'sabbath'. Who is right, you or they?
New Moon literally means first day of the month.
No, it "literally" means, "New Moon". It "signifies" the 'first day of the month' by implication.
And guess what is 14 days later, after a new moon?
I do not have to guess, since I have an exact chronological table from the Bible itself.
What comes 14 days later after the "New Moon", would
depend on how you are counting, exclusively, or inclusively, ordinal, nominal, etc? If you mean the '15th' day of the month (14 days after the "New Moon"), then that day is the "15th" day of the month, and that's it,
unless one of the seasonal feast days lands on it, or if the cycle of the 7th day sabbath from creation (Genesis 2:1-3; Exodus 20:8-11) coincides with the "15th" day of the month. The other option is when the Feast of Tabernacles (Leviticus 23:23-25) takes place, in the 7th month, at the New Moon, being the first day of the month, a yearly sabbath takes place in the same day, and is even possible (but does not have) to coincide with the 7th day the sabbath of God. Consider (assuming 30 day months):
(The Week) (001) (002) (003) (004) (005) (006) (007 Sabbath)
(5-6 Month) (027) (028) (029) (030) (1NM) (002) (003S)
(006 Month) (004) (005) (006) (007) (008) (009) (010S)
(006 Month) (011) (012) (013) (014) (015) (016) (017S)
(006 Month) (018) (019) (020) (021) (022) (023) (024S)
(6-7 Month) (025) (026) (027) (028) (029) (030) (1NMSY) (NM/7Sab./YearS Comb.)
(007 Month) (002) (003) (004) (005) (006) (007) (008S)
(007 Month) (009) (010) (011) (012) (013) (014) (015S)
(007 Month) (016) (017) (018) (019) (020) (021) (022S)
(007 Month) (023) (024) (025) (026) (027) (028) (029S)
(7-8 Month) (030) (1NM) (002) (003) (004) (005) (006S)
(008 Month) (007) (008) (009) (010) (011) (012) (013S)
(008 Month) (014) (015) (016) (017) (018) (019) (020S)
(008 Month) (021) (021) (022) (023) (024) (025) (026S)
(8-9 Month) (027) (028) (029) (030) (1NM) (002) (003S)
That is only when the New Moon takes place at the time that it does, and when the Months are exactly 30 days, rather than 29 or 31 days. The two calculations of time, (weekly) and (monthly) (and so also year), are differing.