I can't respond specifically to Marxist historians of India, but I do have some insight into Marxist historiography more broadly from the 19th century until today - namely, hopefully I can shed some light on your further question of "What is a Marxist historian?"

Between 1880 and 1956, many historians focused on workers’ institutions and political organizations. This focus was driven largely by the notion that the economy could be used as an explanatory mechanism for historical development and for the gains of the ascendant labour movement. Marxist historians, in this period, subscribed to “hard” historical materialism whereby the economic base of a society (meaning the mode of production) determined the development of superstructural societal institutions, laws, and politics. Liberal historians, too, relied heavily upon economic analysis, although they could also be Whiggish in their assessment of labour union developments and political agitation among workers. In 1956, a transitional moment occurred when several British Marxist historians broke from the Communist Party of Great Britain as the result of a number of political and social factors. These historians began to challenge the prevailing economic focus of labour history, highlight the importance of workers’ culture, and offer a new conceptualization of class that prompted a massive theoretical overhaul within the discipline.

The major work within Marxist historiography following 1956, a period known for the emergence of the so-called "New Left," was E.P. Thompson's 1963 The Making of the English Working Class. This book focused extensively on the significance of culture in the development of class-consciousness among English workers in the 18th and 19th centuries. Other historians, such as Herbert Gutman in the United States, revealed the ways in which racial identities contribute to varying experiences of class. These approaches quickly became known as the “new labour history,” and they prompted significant attention to issues of class, race, and gender among Marxist historians during the 1960s and 1970s. Here we see the transition from Marxian focus on institutional labour history to a broader "working-class" history. By the mid-1980s, postmodern and post-structuralist historians began to challenge the validity of categories such as race, class, and gender altogether. This moment of transition corresponded with the decline and fall of the Soviet Union, the emergence and development of neoliberal economic policy in the 1970s and 1980s, declining unionism in the West, and the anti-worker discourses of Thatcherism and Reaganism. Marxist historiography was firmly in crisis by 1990.

In the mid-1980s, postmodernists and post-structuralists such as Joan Scott and Patrick Joyce published screeds against historical practice and methodology, taking close aim at the apparent "metanarrative" approaches of the Marxists. These historians theorized that language was the mechanism through which experience is understood; therefore, they argue, any set of identities – class, racial, or gender – are simply linguistic constructions. Historical materialists took particular umbrage at these theories, as it was felt that the postmodernists sought to deny the political importance of class-based workers’ history. Also in the 1990s, a number of Marxist and former-Marxist historians posited several ways forward for the discipline – Ira Katznelson argued for a renewed focus on institutional and political aspects of working-class life, while others called for new attention to labour internationalism and world-systems theory.

These threads have been followed among Marxist and non-Marxist historians alike since the 1990s. Attention to transnationalism, an expanded view of the international working-class, and the reinvigorated attention to working-class institutions and organizations have all been characteristic of Marxist historiography and labour history since the year 2000. The interpenetration of class, race, and gender, with transnational global experiences of colonialism or capital mobility is of key importance in many works of labour history from this period. The expansion of traditional approaches to labour history corresponds with the emergence of a globalized economy linked by instant forms of communication and a dominant economic and political ideology that highlights consumer culture over working-class identity. At the moment, the discipline continues to struggle with the hegemony of global capitalism, the decline of working-class organizations in the West, and the changing nature of working-class culture in many areas.

Sources:

Leon Fink, “Intellectuals versus ‘Workers’: Academic Requirements and the Creation of Labour History,” in In Search of the Working Class: Essays in American Labor History and Political Culture, ed. Leon Fink (Urbana and Chicago: University of Illinois Press, 1994).

Raphael Samuel, “British Marxist Historians, 1880-1980: Part One,” New Left Review 120 (1980).

Stephen Woodhams, History in the Making: Raymond Williams, Edward Thompson and Radical Intellectuals, 1936-1956 (London: The Merlin Press Ltd., 2001).

• The Aryan invasion debate in India is intensely relevant to the constructions of several very different sets of competing identities:Hindu and Muslim, indigene and foreigner, Aryan and Dravidian, and Hindu communal and Marxist secularist are all permeated in one way or another with a variety of notions connected with Aryan origins.
• There is a tendency to stereotype any local reconsiderations of ancient Indian history whatsoever as nationalist or communal,which is often unfortunate.A wide variety of motives inspire Indian scholars to revisit the topic of Indo-Aryan ori- gins: it is erroneous to lump them all into a simplistic, hastily identified and easily demonized Hindutva category.
• Nationalist historiography is not necessarily bad. It has stimulated asking questions about local cultural and ethnic configurations that would not have occurred to colonially oriented archaeologists. It has brought different assumptions, perspectives,and concerns to the data, exposed colonial predispositions and imperial biases. On the negative side, nationalism has encouraged the misinterpretation of archaeological data for political purposes and ignored important aspects of human history Archaeological interpretation can reinforce and articulate the centralizing policies of emerging nationalisms, as well as be used to legitimize ethnic cleansing or territorial expansion.I'm citing Harke (1991) here.Silberman(1995) also suggests a similar thesis.
• Now, voices challenging the invasion thesis,Indian or foreign,need not be communal,which they unfortunately tend to get stereotyped.This is not to say that there are people who seek this theory for very overt politics.
• To be considered a real Hindu, a person's religious faith must have an indigenous origin.-is the crux of Savarkar's Hindutva.The problem with this seemed to be that it appeared to exclude native Muslims and Christians on the virtue of the fact that their prophets were born on the wrong side of the Indus. This[ideology of Hindutva] has a number of consequences in the way these people have dealt with the Aryan Invasion theory.
• Extreme versions of this have lead to all sorts of nationalistic nosenscical viewpoints,like England was Angulisthana originally or Arabia was Arvasthana,or Muhammad was a Hindu(of the sort produced by PN Oak) to include Muslims as 'Hindus' and thus,Indian citizens.Similar nonsense is spouted by Tamil nationalists as well.
• Hindutva ideology(since I'm talking about it here) can be easily pressed into service in alienating and targeting the Muslims and other minorities in communally volatile, modern-day India.(Hock,1999;Habib;1997,Ananth;1998)
• However,there is a tendency in Western, and in elements of Indian, academic circles to a priori stereotype everyone reconsidering this aspect of Indian history in such ways(like fascists and Nazis:This is particularly strong in places like the universities of Delhi),which has been greatly vexing to scholars like Colin Renfrew, B B Lal,etc. *The point is that: themes resonating with Hindutva, that is, the prioritizing of Hindu culture as the indigenous, and therefore legitimate, heir to hegemonic power in India (with its anti-Muslim subtext), while blatantly and distastefully present in a number of publications, does not pervade the views of all the members of the Indigenous Aryan school.
• You've already mentioned Arun Shourie ,and the controversies in the Indian Council of Historical Research.The "Left-liberal" or "secular Marxist" stereotype is subject to an amount of disgust equal to that of the colonial stooge in Indigenist discourse. Rajaram (1995) states that "in the hands of politically driven historians of post-colonial India, these nineteenth century-creations [viz., arguments supporting the Aryan invasion theory] have become handy tools to be used in support of their vested interest in Marxist ideology and the version of history that goes with it" (xiv). 20 Secular Marxists are accused of maintaining a defunct theory in order to insist that the arrival of the Aryans is analogous to the arrival of the Muslims, Christians, and numerous other groups of newcomers to the subcontinent. In such an amalgamation of immigrants, no one has more claim to indig- enous pedigree or cultural hegemony than anyone else. A secular state, from this perspective, is the only political system that can protect the equal rights of all citizens to define themselves as being Indian with cultural credentials that are as good as anybody else's.
• Thapar's comments are portrayed as "vintage Marxist rhetoric," which has "gratuitously drag[ed] in the bogey of the 'Aryan nation' . . . [as] a blatant attempt aimed at divert- ing attention away from the real issue" (Rajaram 1993, 33). Thapar (1996) finds Rajaram's writings "read rather like nineteenth century tracts but peppered with ref- erences to using the computer so as to suggest scientific objectivity since they claim that it is value-free! Those that question their theories are dismissed as Marxists" (88). From another perspective on the other side of the debate, and in interesting contrast to Thapar's concern about a single "Aryan nation," Talageri (1993) raises the alarm about multiple Marxist mininations: "The first principle of Leftist propaganda is that India is not a nation but a conglomerate of nations. . . . the rationale behind this is that if India breaks up into small "nations," these would be easier for the Leftists to gobble up one by one" .
  Both sides of the debate sometimes refuse to even acknowledge that there is a legitimate controversy,sadly.
• An example of the ridiculousness this reached: The entire proceedings at the plenary session of the prestigious 1994 World Archaeological Congress in Delhi came to a complete (and ridiculous) standstill for more than thirty minutes as "leftist" and "rightist" historians actually clambered onto the dais, physically wrestled each other in attempts to snatch up the microphone, and hurled abuse into the air in front of over two hundred flabbergasted foreign delegates.
• Detractors of the Indigenist school can be just as selective in the views they extract for critique, as the Indigenists they ridicule. Habib (1997) caricatures all the Indigenous Aryanists as believing that the Dravidian language family is not distinct from the Indo-Aryan one—a view held by only a very few individuals and not at all representative of the Indigenous position.Scholars like Satya Swarup Mishra(linguistics at BHU) who abhor Hindutva,face immense frustration at this stereotyping.
• There is no indication that the British consciously exploited the Aryan theory to create a divide-and-rule situation between the North and the South. However, whatever may have been the motives of colonial scholars or missionaries, the dichotomy was most certainly put to political use by separatist voices in the South of India.(Tamil Nationalism,theories of Kumari Kandam,etc).
• Western scholars are often caricatured as discouraging the tribes and lower-ranking castes from identifying with Brahman-dominated Hinduism, gaining their sympathy, and offering them a new set of loyalties in the form of Christianity.
• There are a number of scholars who outline the arguments in favor of the external origin of the Aryans, as well as the arguments that oppose it(in contrast to the shrill and often unpleasant and exclusionary Hindutva polemicists), but leave the issue unresolved—in other words, the Indigenous Aryan position is treated on equal terms with the standard version of events: "European scholars have been trying to prove that the Aryans came to India from outside and their guesses have extended to all sorts of places like Scandinavia, Austria, Czechoslovakia, Russia, Turkey, Central Asia and Armenia. But any of these places cannot yet be said to really be the homeland of the Aryans. We are still in the realm of speculation and are likely to remain so" (Vidyarthi 1970).Also see RC Majumdar.
• A third category of writers gives the Indigenous Aryan school equal and respectful time but passes a verdict against them(Like Kosambi and Luniya)
• A final category—more ideologically sensitive and, as as has been exemplified earlier, usually stereotyped as the Marxist camp—actively opposes the Indigenous Aryan school.
• Ancient Indian pre- and protohistory is an extremely rich and fascinating area of study, but, unfortunately, the origin of the Indo-Aryans has become inextricably enmeshed with the politics of representation. There is every reason to be concerned that if the Vedic Indo-Aryans are interpreted as being indigenous to India, then the Vedic "civilization" and all that developed from it will be construed as "truly Indian" and all subsequent cultural groups known to have immigrated into India could explicitly or implicitly be depicted as "Others."
• If you ask me about what position I take on this stance,I'd be leaning towards Majumdar's position,or a bit more towards Kosambi's That they are open to different interpretations does not at all mean that the Indigenist position can carry the day. But, to my mind, it does mean that an Indigenous position merits a place at the table.

This is from the chapter 'Aryan Origins and Modern Nationalist Dialogue' from Edwin Bryant's 'The quest for the origins of Vedic Culture'.

History is not exact like Physics or Maths, it is open to interpretation. A historian is like a detective who takes (available) evidence in the present time and tries to reconstruct the past. Due to this reason historical studies is open to interpretation.

Indian history is a hotly contested topic between the groups of different ideologies and there is virtually no consensus between these two factions (Left and Right) and yes interpretation of Indian History has deep political implications. (As evident from the Babri Masjid Dispute).

Secondly when one studies history he/she does carries the baggage of an ideological bias, while writing this answer even I am carrying an ideological bias and even you are carrying an ideological bias while asking this question . So in my opinion it would be better for you to enjoy the writings of a historians (as a viewpoint on the events of the distant past) rather than be skeptical about its authenticity due to a certain ideological tilt because history as I have mentioned earlier is open to interpretation.