The question is actually clearly answered by @Tezz. But there is another question from the user @optimus which not only concerns about how the names are derived but also depends on how the calander works.
The Shri-Surya-Siddhanta has a perticular rule to name the month. It places Nakshatras in group of three or two under a perticular month (See table). We name the month Whenever moon becomes full in a nakshatra of that perticular group.
This rule is presented in the 14th chapter of the Surya-Siddhanta.
भचक्रभ्रमणं नित्यं नाक्षत्रं दिनमुच्यते ।
नक्षत्रनाम्ना मासास्तु ज्ञेया: पर्वान्तयोगत:॥ १५ ॥
कार्तिक्यादिषु संयोगे कृत्तिकादि द्वयं द्वयम् ।
अन्त्योपान्त्यौ पञ्चमश्च त्रिधा मासत्रयं स्मृतम् ॥ १६ ॥
The name of the month is to be known by the Nakshatra which is in conjunction with the moon at the time of Purnima-aanta (the end of Purnima). For the month's like Kartika etc. Group together nakshatras in order of twos starting with Krittika, however Group Nakshatras into three's for the Last, second last and fifth month.
||Nakshatras in which Full moon ends
||Purva Phalguni, Uttara Phalguni, Hasta
||Purva Ashadha, Uttara Ashadha
||Shatabhishak, Purva Bhadrapada, Uttara Bhadrapada
||Revati, Ashwini, Bharni
But wait! try checking the calendar for a few months and you will see a violation of the above rule. For example, 8 Nov 2022 The full moon ends with Bharni however the month is named Kartika.
Why? In the translation of Shri-Surya-Siddhanta by Ebenezer Burgess, he writes
...owing to the incommensurability of the times of revolution of the sun and moon, as also to the revolution of the moon's line of apsides, full moon is liable to occur in succession in all the asterisms, and at all points of the zodiac; so that although, at the time when, the system of names for the months originated and established itself, they were doubtless strictly applicable, they would not long continue to be so....
However, today the name of the months entirely dependent upon the solar months. And they no longer regulate according to nakshatra of full moon. In the book, Indian Calender by Robert sewell and Sankara Balkrishna Dikshit, they write
- But the stars or groups of stars which give their names to the months are not at equal distances from one another; and as this circumstance,—together with the phenomenon of the moon's apparent varying daily motion, and the fact that her synodic differs from her sidereal revolution—prevents the moon from becoming full year after year in the same nakshatra...
- It is clear that this practice, though it was natural in its origin and though it was ingeniously modified in later years, must often have occasioned considerable confusion ; and so we find that the months gradually ceased to have their names regulated according to the conjunction of full moons and nakshatras, and were habitually named after the solar months in which they occurred.
The general rule has been that that amanta lunar month in which the Mesha sankranti occurs, is called Chaitra, and the rest in succession.
This is why we don't see the moon to go full in exactly that perticular Nakshatra but near to it. This picture will continue to deviate more and more.
To conclude, the names got their derivation from the Nakshatras. But due to small changes in the motion up in the sky, the names don't align well with the way they are derived.