Let me answer the questions in the order they were presented and offer my general comments alongside and in the end.
Does this site already have a formal definition for the word 'Hindu'
or 'Hinduism?' If yes, what is it? If no, does it need one? If not,
Taking cues from the Islam and Christianity SE sites, we definitely need a "What's on Topic" here page, that is formal and official. Yes, I have looked at the question : Create the on-topic (page) for this community . Whether that is the official, formal and final page is not clear, since it is not part of Help Center. What should go into its contents is better discussed there.
Now, assuming there is a page that defines "what questions can be asked here", there would be no need for defining Hinduism on this site. We should define who can ask and what can be asked. What can be asked includes "about what", which is the religion. We do not have to define the religion itself; for that one must have religious authority. By religious authority I don't mean scholarly authority as a follower which many users of this site may claim, but the authority vested by the religion to define what it is.
We are not here on that job to define a religion. Even if did, religious authority in Hinduism is very fluid and diverse; the closest living authority one can think of, is someone of the stature of the Shankaracharyas. If we are here to evolve a "community definition" of the religion, such a "community definition" in its working terms is already defined by the "What-to-ask" page, we should restrict ourself to that. Like the Christianity SE page says, we don't run the church.
If there isn't a formal definition and we need one, what should it be and why?
As mentioned above, we need to define the site's scope not the religion's scope.
In the interim i.e., in the absence of such a definition, and also if this site fails to reach a consensus on this topic, is it safe to go with definitions along the lines Islam.SE and Christianity.
I find the Islam SE page linked above, near-perfect. It precisely defines what the questioning and answering users should do. It's a beauty that it doesn't even mention Quoran or the Prophet. In this sense, it beats the Christianity SE too. This is to be seen in the context of insistence of the mention of Vedas or Bible by some users. For the amount of controversy, diversity and antagonism Christianity draws as a religion (as much as Hinduism), the Christianity SE page does a pretty good job of drawing the boundaries. An amalgam of the two pages, betterly community-edited on the Hinduism On-Topic page above will perfectly suit our needs.
That said, here is my opinion to some views that have been mentioned in the question, comments or elsewhere, that provide context to the defining exercise.
Moderators claiming to represent Hindus by saying, Hinduism PoV etc :
If this were to happen from a lay user, I would ask the questioner to take it in his stride. After all, every user can't prefix every sentence with an "IMHO and as a lay user" phrase. However, moderators can avoid this by saying, "while not as a moderator but as a lay user view". May be the Topics definition page or the page that defines moderator's roles can mention this. Or we should have a moderator's abbreviation like IMHO. :-)
"what it means to be a Christian or Muslim"
Like the Islam SE page, it's enough that the users identify with Hinduism either as a follower or by expression of interest, even if cursory at that. One doesn't have to go Hindu the whole hog way. I am assuming "what it means" refers to a fair understanding and dutiful adherence, neither of which should be a requisite for a Q & A website. Every religious journey can have curious beginning.
Insistence on the belief in the Vedas :
One of the commenters and one of the answers mention this. I must agree that in the society elsewhere also this is a popularly held view. However, take a look at the following points, with Hinduism in the modern context. From the context, of current followers, prospective followers and interested learners of Hinduism, (in other words, users), who should be the mainstay of the site.
- Thousands of Hindus follow Hindu practices, rituals and worship, without knowing a word of Veda. This can happen in rural, illiterate communities or in urban indifferent communities. Many won't know that such a thing exists, unless a priest or astrologer tells them, even as they stand in front of the deity, listening to those very mantras. Most would have a rough idea of something like "all this comes from Vedas", but won't know that the Vedas are the supreme authority and Veda-Pramana is the final word on everything Even if that is true, like Kumarila Bhatta said. :-)
Should we exclude the lay Hindu who doesn't care about the Vedas, or even if he knows, knows it as a hazy one-word which doesn't have much significance for his daily Hindu practices ? Think of the public school kid who holds his ears when he crosses a Ganesh temple or a rural villager who worships the "mahamaayi thayee" or a "maariyamma" before his farm work, are they less Hindu than (the elite users that) we are ? Do they, or should they be required , to draw from Vedic supremacy ?
To say that the belief in the Vedas (or recognition of its authority) is a must to be a Hindu, we should be able produce the Vedic authority to say so. A reference that says somewhere, "If you don't believe in the Vedas, you are not a Hindu". There may be plenty of references of the consequences of those who don't believe in the Vedas or Veda-Nindhana (say, such as you-will-rot-in-hell), but we should find a reference that insists on belief in the Vedas as a requisite to be a Hindu. Now, from the answers or the comment above, we can, at best, say, it's been a tradition to consider Vedas as a supreme authority, more of a commonplace acceptance. Even if we produce a reference from say, Adi Shankara's works or from the Upanishads, the debate would still be on whether such a reference ABC is a supreme authority, and why not another text or another seer ?.
We must remember that the tradition that we call Hinduism today, might pre-date the evolution of the word itself. Like Judaism, we are a religion with almost no establishable beginning. The word "Sanathana Dharma" refers to this timeless tradition. Hinduism, in a constrained and boundaried way, defined by us within the XYZ scriptures that we think of today, might only be the version that have come down across the ages, alongside other religions that grew in the historical timeline and hence had to acquire a distinct name called Hinduism.
Within its portals, Hinduism ( or more appropriately Sanathana Dharma) , allowed diversity including rejection of God and Hinduism itself. If not anything, the debatory tradition (or Tharka) definitely allowed the discussion of various perspectives, including questioning the authority of what is definite evidence and what isn't. Atheist streams such as Charvaka that definitely reject the Vedas, were allowed to be part of the grand canopy of Hinduism, even as the Vedic priests would denounce these schools and engage in a debate with them. Like Facebook allowing Facebook jokes, LoL. It even allows God to be scolded in its poetry, some verses by Siddhas can mock at Idol Worship worse than atheists can.
For religions that have had their difficulties in interpretation and conflict with modern times, the Islam SE and Christianity SE have done a pretty good job being inclusive, allowing diversity and reducing insistence of an absolute belief from the lay user.
As a religion in practice, Hinduism is vast, diverse and allowed everyone to come, express curiosity, yes question authority for a while, even move away from it and then come back into its fold. The religion allows every person to find his level, and eventually takes them to a level-less realm. The Hinduism SE site should be a reflection of this open pursuit and metaphysical quest that embodies Hinduism. Let's leave it to the curious learner who will step in, learn and hopefully, one day, will accept the Veda-Pramana. Then, he will be a Hindu in his heart, irrespective of geography, SE or the Supreme Court.