In Democracy in America does Tocqueville consider the United States a democracy, or does he have more nuanced views about the United States's form of government?
tl;dr - Yes, he thought the U.S. was a democracy, but he didn't go into a lot of detail about defining what a democracy is. He was part of a new philosophical wave that differentiated the state from the rest of society. His real sophistication was showing how the beliefs and attitudes of citizens would influence their democracy.
Yes, he thought the U.S. was a democracy
de Tocqueville definitely says that the United States is a democracy. He doesn't bury the lead - it's both in the title and is the beginning of the introductory chapter. So what is a democracy? Here's how he describes an ideal democracy:
I can conceive a society in which all men would profess an equal attachment and respect for the laws of which they are the common authors; in which the authority of the State would be respected as necessary, though not as divine; and the loyalty of the subject to its chief magistrate would not be a passion, but a quiet and rational persuasion. Every individual being in the possession of rights which he is sure to retain, a kind of manly reliance and reciprocal courtesy would arise between all classes, alike removed from pride and meanness. The people, well acquainted with its true interests, would allow that in order to profit by the advantages of society it is necessary to satisfy its demands. In this state of things the voluntary association of the citizens might supply the individual exertions of the nobles, and the community would be alike protected from anarchy and from oppression. Introduction
But he isn't really about institutions
The idea of "democracy" is usually thought about in institutional terms: democracies have certain kinds of rules, certain power centers, certain kinds of organization, etc. de Tocqueville's description is different. Although it references some institutional notions, it's really focused on culture. A "democracy" here isn't just about our Constitution, separation of powers, elections, etc. but about how individual citizens think, their values, and eventually their patterns of behavior.
This is where de Tocqueville becomes very sophisticated. It's not that he thinks of democratic institutions in any particular way, but that he thinks of political culture (ideals, values) and institutions (laws, organizations) as being inter-related. The classical view here is that it is the responsibility of the state to develop culture, including political culture - so culture is policy outcome. This line of thinking continued through the Renaissance and the era of social contract theory, in which civil society was thought of in terms of institutions (because the social contract is an institution which binds the state to civil society) Neither IEP nor SEP have articles on civil society, so I'd recommend the Wikipedia history here.
This changed some-time around the 18th century. Starting with Hegel (and later adopted by Marx, de Tocqueville, and many others) civil society and government were viewed as entirely different things. Hegel's definition in Philosophy of Right:
[Members'] association is brought about by their needs, by the legal system — the means to security of person and property — and by an external organisation for attaining their particular and common interests.
Spoiler: The external organization he is talking about is the state. For Hegel, the state was a tool for members of civil society (interest groups) to achieve their ends. de Tocqueville took this one step further by describing how the beliefs and values of those people would characterize the government they created.
The U.S. as a Republic
Drunk Cynic pointed me toward a series of chapters where de Tocqueville describes the United States as a "republic". However, democracies and republics are not inherently different things (see my answer on this question). The author himself describes the U.S. republic in cultural, rather than institutional terms:
What is understood by a republican government in the United States is the slow and quiet action of society upon itself. It is a regular state of things really founded upon the enlightened will of the people. It is a conciliatory government under which resolutions are allowed time to ripen; and in which they are deliberately discussed, and executed with mature judgment. The republicans in the United States set a high value upon morality, respect religious belief, and acknowledge the existence of rights. They profess to think that a people ought to be moral, religious, and temperate, in proportion as it is free.
So although de Tocqueville does describe the U.S. as a democracy, he doesn't differentiate different kinds of regimes in the way that your question suggests. He does describe both a democracy and republic, but he focuses on cultural aspects rather than institutions. This fits the theoretical tradition he is working within.