As Europeans began to explore the world, they encountered people with established religious traditions. The then contemporary understanding of religion was limited to theism and European polytheism. Since both recognize, in some broad sense, the worship of god, this was taken as the basis of religion. All persons are religious, so the theory went, because all believe in God. The problem, according tot his theory, is whether they worship the true God and whether they are saved through Christ or not.
As an example, John Calvin argued that all persons are idol factories in that while all persons believe in God due to an original innate sense, all persons are also fallen and therefore misconceive of God. Benjamin Warfield, in his study of Calvinism, explained Calvin's theory of religion which has been extremely influential down to the present through thinkers like Friedrich Schleiermacher and Rudolf Otto.
"In the first of these aspects, as we have already pointed out, religion is with him the vital effect of the knowledge of God in the human soul; so that inevitably religions will differ as the conceptions of God determining our thought and feeling and directing our life differ. In the estate of purity, the knowledge of God produces reverence and trust: and the religion of sinless man will therefore exhibit no other traits but trust and love. In sinful man, the same knowledge of God must produce, rather, a reaction of fear and hate -- until the grace of God intervenes with a message of mercy. Sinful man cannot be trusted, therefore, to form his own religion for himself, but must in all his religious functioning place himself unreservedly under the direction of God in His gracious revelation. In its second aspect, then, we perceive Calvin carefully framing his definition so as to exclude all 'will-worship' and to prepare thew ay for the condemnation of the 'formal worship' and 'ostentation in ceremonies' which had become prevalent in the old Church. The position he takes up here is essentially that which has come down to us under the name of 'the Puritan principle.' Religion consists, of course, not in the externalities of worship, but in faith, united with a serious fear of God, and a willing reverence. But its external expression in worship is not therefore unimportant, but is to be strictly confined to what is prescribed by God: to 'legitimate worship, agreeable in the injunction of the law'" (B.B. Warfield, Calvin and Calvinism, 38-39).
There is much that is helpful here and I can only consider this passage from one perspective in this post. I think Calvin can be taken in the way of defining religion I'll suggest below, but that he is often taken to mean everyone knows God and this is the beginning of religion. What must be done is rekindle this awareness, rather than dealing with answering basic questions.
This definition of religion treats religion as more and less pure in its conception and worship of God. A person is therefore either religious or not religious, and if religious then either more or less pure in worship. The consequence was that as Europeans encountered persons from around the world, the assumption was that they were somewhere on this continuum with respect to religion. Since they were not atheists they were religious (and therefore believed in God in some sense), and the problem was a matter of correcting errors about God and worship and in offering salvation through special revelation.
By way of contrast, we find that there are examples of what seem to be religious groups and yet are not theistic. Rather than there being "religion," there are "religions." We need to define these formally so that we find what makes them similar in the face of their dissimilarities. It cannot be belief in God, or use of scripture, or rituals, since not all religions have these.
What these do share in common is that they all give answers to the most basic questions that can be asked, and use these answers to interpret the experiences of life. So we can define religion as the set of basic beliefs used to interpret experience. While not all religions believe in God (as defined in theism), all religions answer the question "what has existed from eternity?". While not all religions have scriptures or a hierarchical leadership, all religions answer the question "what is the highest authority?". And finally, while not all religions affirm the need for redemption, all answer the question "what is the highest value?".
There are implications of this definition. The definition considered from Calvin allows that there are perhaps some that are atheists and therefore not religious. The definition I've given implies that all persons (even atheists) are religious in that all persons answer basic questions and use these answers to interpret their lives. This is a formal truth about people and how thinking works.
However, it also follows that people are more and less consistent in their awareness and thinking about basic questions and the answers that influence their lives. As we look at how someone lives we can discern the answers by which they are living, and yet when we talk with the person it often becomes clear that he/she is unaware of these questions and basic beliefs.
And yet these answers are beliefs, which means they are cognitive and therefore either true/false. Religion is not fundamentally mystical or experiential, but is fundamentally giving answers to basic questions. This is true of those who emphasize the mystical or experiential. All experiences must be interpreted including mystical experiences. The manner of interpretation will differ depending on the basic beliefs of the interpreter.
Finally, religion is not a matter of blind belief. When I see movies or television shows that are promoting religion, they invariably do so by promoting the need for blind belief in some areas of life. They think this is doing religion a favor. However, this is accepting the dichotomy between faith/reason. By way of contrast, I'd suggest blind faith is not the same as faith itself. When persons have faith, they believe some proposition such as "God is one being and three persons." They are able to have faith only to the extent that they understand what this means. Therefore, a trial of faith is a trial of understanding. The primary question is not whether a faith commitment is true/false, but what does it mean? If it is meaningless then it cannot be an object of faith. We cannot say "I have faith so that I might understand," but "I have faith to the extent that I understand."
This definition of religion and these implications allow us to formally compare answers to basic questions given by the various religions of world history. Other definitions that presume all persons believe in God actually hinder such comparisons and keep us from studying basic beliefs by shrugging them off as unimportant since everyone already knows. The reality of the matter is that people do not know, and history is full of conflict between groups with competing answers to these questions.