One of the central claims made in this course is that religious fundamentalisms are reform movements, but that there are also what might be called liberal reform movements operating simultaneously within a religious tradition. Since both the fundamentalist approach and the liberal reform approach are involved in bringing about change within their religious tradition, it can be difficult at times to distinguish between the two. For guidance on general principles in making this distinction, look at the course packet, pp 35 - 48.
It is especially important to read those pages carefully before attempting to make the fundamentalist/liberal reform distinction in connection with Hindu reform movements. Both the Hindutva movement and various more liberal reform movements will pop up on the Web if you search for Hindu reform pages. This happens because both the Hindutva movement and the more liberal movements all openly see themselves as reform movements (unlike the case in Christianity, for example, where it is not so common to recognize that fundamentalists are actually trying to be reformers, and where it is not so common for the fundamentalists to refer to themselves as reformers).
Liberal reform in Hinduism in response to modernity has been in process for well over a century -- some might say longer.
One aspect of this type of reform has to do with attempting to reshape the Hindu spiritual vision in the light of ideas originating in the West. Some key figures you can look up on the Web in this regard are Ram Mohan Roy, Keshub Sen, and Ramakrishna -- all of whom became famous throughout India for suggesting new ways of thinking about Hindu theology. For example, a brief description of Roy's and Sen's activities is found at the Encyclopedia Britannica site here. You can read about Ramakrishna's movement here. In this interchange with the West (triggered largely by the Raj), monotheistic ideas that were already present in the Hindu tradition from of old received new emphasis and were consciously developed, and the caste system came under attack on moral grounds (which meant that the whole conception of rebirth and karma in relation to caste had to be reconfigured somehow). So, if you visit the Ramakrishna site above, you should pay attention to what is not said, as well as to what is said. (E.g., some absent things: multiple gods, karma, rebirth). Another similar famous figure you could look for is Tagore.
But there is another kind of liberal reform in Hinduism, a more political kind. Bal Tilak and Mahatma Ghandi are famous Indians who are perhaps seen by outsiders simply as political figures instrumental in the Indian independence movement to establish India as a modern nation-state independent of foreign domination. However, these men saw their political lives as embodiments of Hindu spirituality, and they represent a modern liberal view of what Hinduism can mean today in political culture. In other words, if one looks at the political ideals of these men, one finds a kind of meld between politics and religion (similar to what one finds in activist fundamentalist movements), but it does not have the flavor of fundamentalism (in my judgment, anyway), but rather the flavor of liberal reform -- changing the picture of what Hinduism means in the modern political world. You might look at this paper on Ghandi's religious philosophy for some information about this topic.
In distinguishing Hindu fundamentalist reform from liberal reform, it is easy to be baffled by the common occurrence of the idea that all religious traditions in India are legitimate paths to spiritual insight, or paths to the divine. All the Hindu reform movements talk that way, whether they are fundamentalist in character or more liberal. So, if there is such a thing as Hindu fundamentalism, it is not marked by the exactly same sort of "We have the truth and everyone else is wrong" stance that fundamentalisms in other traditions display. And thus, one cannot distinguish fundamentalism in Hindu life from liberal reform by means of this particular feature of fundamentalism, when taken in the usual way. However, I think that if you look a bit deeper, you can find evidence that there really is nevertheless a way that Hindutva thinking does turn out to produce a kind of "we/they" way of talking, in which people who disagree with the Hindutva agenda are labeled as bad Hindus, or sleeping Hindus, or even traitors to Hinduism. I don't see this same element in the liberal reformers.
It is also possible to distinguish Hindutva thinking from liberal reform thinking by looking at other aspects of what typically distinguishes fundamentalisms from liberal reform movements (as described in the packet). If you were to go through the relevant sections of 35 - 48 you would find some things to look for in these movements in order to distinguish them.