Why Ancient Year was called Varsha

Rig Veda mentions 3 seasons (Summer, Rainy and Winter) which are observed only in India and neighbouring countries.
In most parts of the world, rain comes either in Summer or in Winter (in few places as snow fall).
Ancient Year Varsha Rainy Season

Rig Veda (6-32-5) observes that every time Sun tilted from North towards South, it rained.
This rain (Varsha in sanskrit) started every 12 months around same time and year was counted from that date.
This is the reason why year was called Varsha in vedic age.
As agriculture was main profession in India, rains were given more importance and calendar (Panchang) was formed around it.
Later Ayanas were divided according to tilt of Sun towards North (Uttara-Ayana) and South (Dakshina-Ayana), making 2 parts of a Varsha (year).
Here, Ayana (travel/movement) of Sun is considered.
Rig Veda also mentions that people began their new year with rainy season.
Rig Veda (2-24-5) states : “The ancient verses of Brahmanaspati opens the door of future waters after months or years. Therefore without any effort, the heaven and earth enjoy each others waters.
Ancient Year starts with Rainy Season

Cycle of water known to ancient Indians

Rig Veda (1-64-51) states : “This water is common. It goes up and comes down. Rains saisfy the earth, while Agni satisfies the sky.”
This verse clearly shows that ancient people knew about water on earth is being recycled through Sun rays as clouds and rains.
Rig Veda (7-103-1) mentions : “Like a brahmana observing a vrata (religious act), frogs which were sleeping for a year, began croacking which fascinated the rains.”
Rig Veda (7-103-7) states : “When rainy season arrives, on that first day of the year, frogs appear everywhere.“.
Frogs in rain Rig Veda
Rig Veda (7-103-8) states : “Frogs croacking like a brahmin, who sings yearly Psalms. Frogs soaked in sweat come out in the open ground as soon as rain appears.”
All these verses prove that at the end of summer, rain begin which dates to Summer solstice (21-22 June), is when ancient year (Varsha) began.
Rig Veda (7-103-9) mentions that at the end of year (end of summer), there was extreme heat and with arrival of rains, people celebrated new year.
The tradition of performing marriages to frogs in order to bring rains must have started around that time and still continues in many areas.
Ancient Year started with Agriculture

Rig Veda composed around 23720 BCE, praises only rainy season but not summer or winter.
Above verses were composed by Sage Vasishta, whose 100 sons were killed by King Kalmashapaada (mentioned in Rig Veda 7-104).
This king Kalmashapaada was 15 generations earlier than Lord Rama.
So the tradition of celebrating new year at start of rainy season and called an year as Varsha, started much before Ramayana.
Even Valmiki Ramayana mentions year as Varsha, along with 3 seasons.

Valmiki indirectly mentions timing of new year (varsha) in Ramayana, Ayodhya Kanda (77-25) as

उत्थितौ तौ नर व्याघ्रौ प्रकाशेते यशस्विनौ || २-७७-२५
वर्ष आतप परिक्लिन्नौ पृथग् इन्द्र ध्वजाव् इव |

Translation : The two illustrious tigers among men (Bharata and Satrughna) who had got up, appeared like two banners severally raised in honor of Indra (the god of celestials) soiled through rain and the sun.

Rain and Sun (heat) come together in one month during June (summer solstice). Banners of Flags were raised in honor of Indra (rain god) just before rains started and they were defaced by getting exposed to heat and rain in same month.
Rigveda 1-161-13 states : “Who awakened Rubhus ?” (Rubhus means clouds)
The Sun replied, ”The dog, because today is the end of the year.” . The dog means the Canis Major or the Mrigasira Nakshatra.
End of year can be considered as end of a tough summer season and time for agriculture (rainy season).
Clouds were awakened by the Dog. It means that the clouds began rainy season when sun enters Mrigasira Nakshatra.
In present era, rains start in india when sun enters Canis Major (Mrigasira Nakshatra) in Niryanaya method.
When Sun enters Arudra Nakshatra (13 days after Mrigasira), earth is said to menstruating. Which means that the soil is soaked with rain water and is now ready to grow plants. This is when farmers sow seeds.
Ancient Year started with sowing seeds
The traditional Puri Jagannath Rath Yatra also starts after rainy season commences in India.
It is said that the wheels of Lord Jagannatha start moving at that time (begining of year cycle).
Ancient Year started with Puri Jagannadh Rath Yatra

Taittariya Samhita (7-4-8) mentions different year cycle. It says that Uttara Phalguni was first night of a year and Poorva Phalguni was last night of previous year.
Satapatha (6-2-2-18), Gopatha (6-19), Pancha Vimsa (5-9-9) and Sankhyayana Brahmanas support above theory.
Taittariya Samhita is part of Yajruveda, which was composed much later and if New Year began at Uttara Phalguni, then Sun must be diagnally in Uttara Bhadrapada, which is modern day Vasanta Ritu or Vernal Equinox around 21 March.
So, the tradition of celebrating new year (Ugadi) started in March-April, started much later but both methods were practiced simultaneously as Yajurveda was made more popular in South India (Krishna Yajurveda).
This is why Taittariya Samhita (6-5-3) poses a question : “The vessel of Ritus has two mouths. Who knows the mouth of ritus ?.

The idea of Ugadi (new year) starting in March-April comes from the celebrations of Devi Navaratri during October.
It is said that all the warfare done by Goddess as described in Devi Bhagavatam was done around midnight (12 AM).
Goddess killed demons at Midnight
When an year of human cycle is equivalent to one day of cycle for gods, each month lasts for about 2 hours.
So the gods must be awake after 4 AM (in mid december) and that is when Dhanur-masam is celebrated by Vaishnavas.
Their day starts around 6 AM (in mid January for humans) and that is when Makara-Sankranti and Uttararayana start.
Between 10 am – 12 pm of gods (Sun) time is when they are most active and ready to communicate with humans. This is when Sun starts getting hotter in India and absorbs water from earth.
By their evening time (between 4-6P PM) sun cools down and starts showering back water as rains.

Contrary to popular belief, concept of calling present Indian subcontinent as Bharat Varsha started much before Ramayana, where people believed that Indian subcontinent was the land of rising sun.
Bharat name as nothing to do with Emperor Bharata.
Sugreeva in Kishkinda Kanda mentions that Jambu Dwipa, the subcontinent lies southerly to that heavenly Mount Meru, and down East-South of Saumanasa, this subcontinent can happily see the First Sun, daily. This gives raise to the Indian name of India, that is Bhaarata, bhaa rataH = Sun delights…’ The place where the Sun delights firstly, that is bhaarata…

Bharata Varsha name
Also Goddess Saraswati is known as Bharati (One who dwells in light/knowledge) and when vedas were composed on banks of saraswati river, rishis called the people living in this country as sons of Bharati and the land as Bharat (where people dwell in knowledge).
Land showered (varsha or rained) by blessings of Bharati (Sararwati) was called as Bharata Varsha.