Hindu is from Persian for Indian. The land, climate and history are crucial.
The earliest elements of Hinduism are called Protohistoric: 4000 BCE and on to 2200. This included Yogic posture male god, female goddesses, phallic symbols, animals, trees, water and purification.
The Vedic period up to 500 BCE involved the Rig-Veda of hymns to divine powers (devas) and fire sacrifice and a sacred plant, Soma, to make a spiritual drink. There was a concern with magic and the correct performance of ritual. Upanishads moved away from this towards personal encountering of the one God, The Atman, self, goes with Brahman. Samsara doctrine appears which promises endless rebirth until unity with Brahman, each birth determined by Karma.
500 BCE-500 CE saw the Classic period which emphasised living in this world. There were the beginnings of caste like social structure. Gods appeared and epics Ramayana and Mahabharata (containing the poem Bhagavad Gita with Krishna and loving devotion to the Lord) emerged. Vedic sacrifice declined.
500 CE to 1800's CE. Middle or Mediaeval Hinduism took place with Buddhism becoming separate. Caste hierarchy grew. A view that Brahman (God) is real, all else (world, individual, even devas) is unreal. Some rejected Hindu and Muslim exterior religion for a personal God without attributes. One of these was Guru Nanak leading to Sikhism.
So to the Modern period. It was the period of colonialism. When the British arrived Hinduism was stagnant. Many responded positively and negatively to Western influences.
Ram Mohan Roy (1772-1833) studied Upanishads and rejected idols. A Rationalist and he led to the end of widows killing themselves on husband's funeral pyre. Founded Brahma Samaj.
Debendranath Tagore advanced the primacy of reason and conscience over the Vedic scriptures. A pro-Western movement rejected caste and improved the status of women. There was great social reform and child marriage became illegal. At the same time some rejected the West and returned to the Vedas.
Vivikenanda attended the World Parliament of religions in 1893 and said all religions are true, Hinduism is the mother of them and India is spiritual and the West materialistic, though he wanted Western science.
Mahatma Gandhi promoted satyagraha or truth and ahimsa, non-violence and called the untouchables Harijans, the people of God. He helped to remove the British and India became a secular state. Hinduism thus was assertive and reborn.
From my own teaching practice notes