Abrahamic religions and Buddhism
It's a religion because it jointly stand for the concepts like theory of Karma, transmigration of soul, concept of heavens and hell, the inherent unity of self and Supreme and the ultimate mission to merge back into the Supreme and become one with God - i.e. to achieve Moksha.
These are concepts that are unique to Hinduism and all the various branches you referred too, including the agnostic Buddhism and atheistic Jainism. It gives this tradition a unique color and flavor that is uniting in nature making it ONE COMPLETE RELIGION on its own.
It's a way of life, because on the path to Godhood, it gives liberty to its adherents to practice the path they feel is more in tune to their respective nature.
If they are theistic in nature, and loving at heart, they are welcome to follow the way of Bhakti (devotion) and sing and dance their way to liberation.
If they are doubtful in nature, they are allowed to use their doubt as a tool and seek out the truth following the athestic / agnostic Shramana Sutras.
And that's the whole beauty of Hinduism. And also it's greatest strength - The acceptance of the fact that there could be more than one path to achieve the ultimate truth.
And this is also what makes this tradition deeply secular. So much so that it can see the essence of divinity even in the people that doesn't belong to their tradition.
After all liberation, (unlike the belief of balance all the religions of this world); is not something that is sole right of any one religion. For if there Is God, then he would be more pleased to see rightful people following any path, rather than seeing people following any of His assumed preferred path but doing the wrong things.
It's not just the latest and the purest stream of Hinduism, but also it's greatest strength. After all, their very genesis was in the vow to protect Hinduism when fanatic Muslims like Aurangzeb were hell bent upon destroying it.
If it was not for the sacrifice of our Sikh brothers, then probably we would not have been discussing this question today. For Hinduism might have been wiped off like all the other religions from Turkey to Afghanistan.
Sikhs are not just the protectors of Hinduism, but also inheritors of Raja-yoga tradition of its age old Vedic legacy, which makes a person both a soldier and a saint at the same time.
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