Weber's distinction is well-known from his lecture "Politics as a Vocation". He identifies:
Charismatic authority: authority attached to a specific individual and the perception of their exceptional characteristics, transcending normal social boundaries (e.g. Hitler, Napoleon, Nader Shah, Chinggis Khaan, Muhammad, Caesar...). This authority dies with the individual, but can serve as the basis of...
Traditional authority: authority asserted on the basis of tradition, custom, and habit. Weber notes: "This is 'traditional' domination exercised by the patriarch and the patrimonial prince of yore." And he distinguishes this from...
Legal authority: authority asserted on the basis of established rules; again Weber: " by virtue of the belief in the validity of legal statute and functional 'competence' based on rationally created rules. In this case, obedience is expected in discharging statutory obligations. This is domination as exercised by the modern 'servant of the state' and by all those bearers of power who in this respect resemble him."
The last one is the one I have the most issue with, partly due to the prima facie issue of presentism and the desire to distinguish his contemporary society from those of the past. Weber basically identifies the issue I have himself without actually adressing it, in the section where he introduces the professional administrator personally dependent on the monarch:
These are propertyless strata having no social honor of their own; materially, they are completely chained to him and are not backed up by any competing power of their own. All forms of patriarchal and patrimonial domination, Sultanist despotism, and bureaucratic states belong to this latter type. The bureaucratic state order is especially important; in its most rational development, it is precisely characteristic of the modern state.
Here he introduces the factor of "rationality" to distinguish the modern state from what he calls "Sultanist despotism" (presumably meaning administration by eunuchs and the like). He comes back to this idea later:
A fifth stratum [of "politically exploitable strata"], the university-trained jurist, is peculiar to the Occident, especially to the European continent, and has been of decisive significance for the Continent's whole political structure. The tremendous after-effect of Roman law, as transformed by the late Roman bureaucratic state, stands out in nothing more clearly than the fact that everywhere the revolution of political management in the direction of the evolving rational state has been borne by trained jurists. This also occurred in England, although there the great national guilds of jurists hindered the reception of Roman law. There is no analogy to this process to be found in any area of the world.
His assertion that this class is unique to the Occident is clearly not true, but he glosses over the obvious counterexamples:
All beginnings of rational juristic thinking in the Indian Mimamsa School and all further cultivation of the ancient juristic thinking in Islam have been unable to prevent the idea of rational law from being overgrown by theological forms of thought. Above all, legal trial procedure has not been fully rationalized in the cases of India and of Islamism. Such rationalization has been brought about on the Continent only through the borrowing of ancient Roman jurisprudence by the Italian jurists. Roman jurisprudence is the product of a political structure arising from the city state to world domination--a product of quite unique nature. The usus modernus of the late medieval pandect jurists and canonists was blended with theories of natural law, which were born from juristic and Christian thought and which were later secularized...
His third category seems entirely dependent on a priori accepting his peculiar notion of "rationality" which he nowhere tries to define, but which I suppose is probably elaborated on in his other works. Regarding the validity of "rationality" as singular, unique and exceptional in the particular instantiation of his time and place seems to be the only thing separating his "legal authority" from "traditional authority". This seems like a departure from the far more universal category of "charisma" and "tradition" and would essentially be replaced by whatever mode of thought was preferred by the particular civilization using his categories (e.g. "Shari'a authority", "Confucian authority", "Dharmic authority"...)
Or am I completely off base with this? Precisely because his distinction between charismatic and traditional authority seems so useful and is so often referred to his third category irks me. Have there been any attempts to revise this system while retaining its relative simplicity? A more useful distinction seems to be whether authority is exercised by professional civil servants or only by e.g. landed nobility who themselves possess traditional authority.
I vastly prefer Weber's presentation of this theme in Chapter 3 of Economy and Society.
After defining what he here calls the "Three Pure Types of Authority," he differentiates them thus:
In the case of legal authority, obedience is owed to the legally established impersonal order. It extends to the person exercising the authority of office under ti by virtue of the formal legality of heir commands and only within the scope of authority of the office. In the case of tradition authority, obedience is owed to the person of the chief who occupies the traditionally sanctioned position
The italics are Weber's own. Legal authority is grounded in an established system of rules that exists outside the individuals enacting the authority. Traditional authority does have some systematic nature, but authority ultimately still rests with the individual. Charismatic authority lies entirely with the individual.
Meaningfully distinguishing traditional and legal authority is a problem Weber foresaw. He later admits that these three forms are not distinct or exclusive:
In general, it should be kept clearly in mind that the basis of every authority, and correspondingly of every kind of willingness to obey, is a belief, a belief by virute of which persons exercising authority are lent prestige. The composition of this belief is seldom altogether simple. In the case of "legal authority," it is never purely legal. The belief in legality comes to be established and habitual, and this means it is partly traditional.
For as much time as spends on typologies in the chapter, Weber's theoretical discussion in section 13 of the chapter is firmly ambivalent about the utility of any of those types. To me, the utility of the "three pure types" is not in the forms themselves, but in their multiplicity: legitimacy and power and questions will several answers, and no one strategy will work to govern all people.
You're right that much of the reasoning behind a separate "legal" category relies on other works. Legal, bureaucratic authority is just one aspect of the suite of modern ideals. Weber's idea of the modern is most typified by "disenchantment," the move away from religion and 19th-century Romanticism, and "rationality," the move towards defined, calculated goals that pervade society. (I like to think of neoliberalism as the apex of Weberian rationality) Legal authority fits in with these themes of modernity. Its traditions are disenchanted, its charisma is rationally optimized.
You're also right that the undertone of "what makes my era special" is problematic. It's a common trope of social theorists at the turn of the century, but not a question that could really be answered for a few decades more.
That said, Weber's idea of leaders constantly, actively vying for legitimization is an enduring theoretical direction. In a way, it predates the ideas of Bourdieu and other mid-/late-20th century theorists who saw most every social category as the result of human action.
For archaeologists, legitimization is a powerful theoretical lens because its much more visible in the ground than vague concepts like "status." In Andean archaeology, there are several ongoing debates on whether we should call certain states empires. One approach to an answer is to look material remains comparable to obvious empires (e.g. Rome, the Inca). Another approach is to go back to not what an empire is but what it does, i.e., how does an empire legitimize itself. If we understand legitimacy as a constant problem rulers must face, and one with multiple solutions, we broaden our understanding of what might constitute an empire. For those of us in a second camp, an empire is simply an expansionist state that encompasses several regions and ethnicities. Weber tells us that state can legitimize itself in multiple ways, and can thus have multiple patterns of material remains.