The word Vaidῑkam means that which follows the Vedas. All branches of Hinduism claim their roots to the Vedas. Therefore, they call themselves Vedic religions (Vaidῑkam). But they are not, as defined by Kạnchi Periyavạl. He says that a Vedic religion has to accept all portions of the Vedas and not oppose any part of them. They should also follow all the 40 samskạrạs mentioned in the Smriti texts and not add new ones. They should follow all the varnạṣrama darmas, not just portions of them.
For example, some focus on only one aspect of the trinity, such as ṣiva (pạśupáṭam) or Viṣnu (pạncarạtram). They tend to exclude other aspects of the trinity. They add samskạrạs not mentioned in the smritis, such as placing mudras on the body as in some sects of Vaiṣnavaites. Some of them also promote the idea that the authority they derive is not only from the Vedạs but also from the ạgamạs and purạnạs. They also tend to claim that their method of worship was given to them by their favorite deity. They become separate sects when combined with tạntric rituals.
There are also the straight tạntric religions such as that of kulạrnava sạktam. In general, Tạntric religions 1. Claim independent authority apart from the Vedas. They have their own “purạnạs” which they claim to be revealed texts. 2. Claim exclusive loyalty and have initiation rites. They may even condemn other Gods. 3. Follow separate rituals not given in the Vedạs. 4. Some follow esoteric and extreme practices such as the five “M” s (the five Ms are: madya (wine), matsya (fish), mạmsa (meat), mudrạ (hand symbols or dried grains) and maituna (sexual union).
It appears that there were over 70 religions in India at one time, most of them calling themselves as Vedic religions. SAt times past, some of them practiced very crude rituals such as animal and human sacrifice. Adi Sankara is credited with eliminating many of these sects and consolidating the rest into five major branches of theistic Hindu religion – Ganapatyam (of Ganapti, or Ganesha), Saivam (of Shiva), Vaishnavam (of Vishnu), Sauryam (Of Surya or the Sun) and Saktam (of Shakti or Mother Goddess). One caveat is that neither the word Hindu nor the word religion (as used in English) were known or used at that time. The name was “Sanatana Dharma”.