Month names come from three sources as I have seen so far.
candra-mAsa names come from the nakSatram near which the full moon most often occurs. For eg this is the śrāvaNa mAsa in the amāvāsya-anta system followed in Andhra, karNATaka and mahArASTra (among other states). So the full moon occurs around the śravaNa nakSatram in the makara rAśi. It’s neighbouring nakSatrANi are uttarASADa and dhaniSTa (śraviSTa).
We immediately expect the Sun to be in the vicinity of the opposite nakSatram and rAśi.
ravi-arka names are derived from the Hindu Sidereal Zodiac signs that the Sun is in. Thus the ravi-arka names are related to the chandra-mAsa names approximately.
The Rtu: are seasons that directly depend on the the position of the Sun. For example, the old names of the two mAsa: of vasanta or spring season are madhu and mAdhava. (We have 6 seasons).
We know that spring occurs around the spring equinox, which is when the full moon is around citra nakSatram and sUrya is in the vicinity aświni, ie caitra-mAsa, ravi-arka. Thus caitra and vaiśakha are the new names spring months.
And thus, with some corrections, this pattern is maintained. My father said that adhika (extra) mAsA: (like leap days) are those in which the sun does not change his rAśi. ie ravi-arka remains the same for both the adhika and the regular candra-mAsa.
And what does the tithi tell us?
A tithi is about a 12 degree separation between the Sun and Moon and a nakSatram is a 13.33 degree duration. In one tithi, the Moon has moved ahead of the Sun by about a nakSatram, but the Sun too has moved about a degree. So we have this 12 degree separation.
If you know the mAsa name and the tithi, you have a fair idea of which nakSatram , the Sun and Moon are near without any major calculations. And more exactly with calculations and corrections.
It is fair to say that mAsa and tithi contain nakSatram locations of the Sun and Moon too. And the inner planets particularly Mercury (budha) are never too far from the Sun.