Time measurement in ASOIAF

According to the books and GRRM months don't have names in the world of Ice and Fire, but can be referred to as such. More commonly are they referred to as "moons" and moon turns. Maesters say in example: "The twenty-second day of the fifth moon of the year 130 AC."

Now there is one thing that is changed in the later books: AL (After the Landing) is more commonly referred to as AC (After the Conquest) now (and the time before BC, meaning Before the Conquest. But due to game limitations this is unimportant to the mod). This was due to Aegon's Landing being two years prior to the actual beginning of the measurement at the end of the conquest.

I wanted to point out wouldn't it be more true to the lore if the mod were to use these names for months instead of our world's names and AC instead of AL?

• First Moon
• Second Moon
• Third Moon
• Fourth Moon
• Fifth Moon
• Sixth Moon
• Seventh Moon
• Eighth Moon
• Ninth Moon
• Tenth Moon
• Eleventh Moon
• Twelvth Moon

GRRM also confirmed that months and measurement, albeit mostly nameless, are the same as in our world.

If this has been changed already (playing on an older version because of my saves), I want to apologize for the unneccessary inconvienience. I just thought this would add on to the immersion.

Edit:
For reference here is the link to the wiki page: awoiaf.westeros.org/index.php/Measurement

Also, I posted this in the wrong thread before and apologize for posting the same again.