IMO
The word Hindu/Hinduism is very non specific and can mean different things at different times depending on the person using it and the topic of discussion and the person's intention etc. This problem is used by Brahminists to divert the topic or try to confuse people by claiming many contradictory things. So Brahminism is (relatively) more specific term when discussing Indian religious and social matters.
Two most important features of Brahminism I can think of are:
(1) believing in /claiming supremacy of Vedas - (which indirectly means supremacy of Brahmins since Brahmins have always had copyright on Vedas).
(2) believing/promoting/justifying caste system. (which since last few decades is difficult to do openly so they do it indirectly or in private or on anonymous internet forums)
Vedic religion could have been used as an alternate term but even that would complicate the matter and give chance to Brahminists to divert discussion. because firstly, it does not matter what the books say. Since only Brahmins had the right to interpret Vedas, it was not a religion based on what Vedas say, but based on whatever the Brahmins think/claim about what Vedas say. Then to complicate the matter, that religion of Yadnyas and sacrifices for selfish demigods almost died after Non-Vedic Sramanic religions started to dominate; then in it's Mark II avatar the Sanatana Dharma was mostly about the pathetic storybooks of Puranas, various rulebooks, rituals all written by different Brahmins as per their fancies and convenience, to maintain caste system and concentration of power (knowledge, weapons, wealth) in hands of few dominant clans (upper castes) . All even before Muslim invaders arrived.
All this is too confusing for us common folk and every person can not be an expert on history and religious scriptures. So to keep matters simple, I would rather prefer to use some loose term for people who try to promote or are apologists of this system characterized by 1.Caste system, 2.supremacy of Vedas (and justify other Brahminical Shastras). I think everyone understands what the word means and more than exact definition and wordplay, what matters more is real intention.
IIRC, the word Brahminism is not a new invention, there have been many parallel religious movements/sects in history and they have used similar words to try to distance themselves from the dominant Brahminist system.
1. You have already made it clear that there is no clear definition of Hinduism.
2. Brahmins are a CASTE of Hinduism and are clearly defined as those pursuing attainment and giving knowledge.
3. Care to tell us which caste Manu, the original law giver, belonged to? Was he a Brahmin? If so, how come his sons funded Suryavamsha of Rama and Chandravamsha of Kurus? Was Ram a brahmin or a kshitya?
4. Assuming Manu was a Brahmin, which caste he uphold as supreme in following laws:
i. Pregnant women are to ferried free of tolls?
ii. Physicians are unqualified to judges, but as witnesses they are to be called only as a last resort. But simultaneously laid down that a shudra is a competent witness oe even can be member of council of ministers?
5. Which caste Sushruta, the founder of surgery belong. If you don't know it was Brahmin. Which caste did he hold supreme when he advised shunning socialising with physicians. Again, which caste was he upholding when he laid down that abortion is to performed only on medical advice for saving the life of mother? Is it any different from modern laws?
6. Do only Brahmins believe in Vedas? Come on, which Hindu does not perform marriages according to Vedic rituals? Which Hindu, non Brahmins, call a maulvi or bishop for performing marriages?
7. If Brahmins wanted to concentrate wealth why this adage: Brahmin ka dhan keval bhiksha. Complete is :Foolish wife, others want wealth, but Brahmins can subsist on alms.
8. By sramnic thing you perhaps mean Buddhism [damned if are even clear about it]. Don't you know that after about 200 BCE, Buddhism declined steadily and by about 800 CE it was finished in India? Jainism never had many adherents anyway.
I have known a few anti brahmanists, who roundly cricise Brahmins. Of course without basis. When asked who officiates on their weddings, maulvis or pandits, you can know the answer without telling by me.
Using the term Brahmanism is just a wordplay to denigrate and attempt to ostracise Brahmins from the Hindu fold.
Btw, Brahmins need not and do not justify caste system. It ia admirably done by those who carry caste certificates and cling to their castes.
PS: Don't tell us what you think, tell clearly What is Brahmanism and a Brahmainist.