What kind of democracy are the anti-federalists envisioning and is it compatible with the federalist conception of democracy?

First, it should be acknowledged that neither the Anti-Federalists nor the Federalists envisioned a democracy, at least not in the current sense of the word. Whatever their opinions of the Constitution, most early American political leaders favored a republic, a government that filtered the will of the people through layers of representation. Federalist leaders in particular were very uneasy about popular government. That being said, the Anti-Federalists were concerned about the powers that were vested in the national government under the new Constitution. They thought that the states should maintain most of the political power, though some of them admitted that a new government was necessary. Most of the "Antis" advocated the establishment of amendments that would better protect the individual rights of the people and limit the power of the federal government in favor of the states. The Federalists, on the other hand, argued that the rights of the people would be adequately protected by the central government under the Constitution. They believed, in fact, that the democratic excesses in many state constitutions would lead to civil discord and despotism. So the Anti-Federalists could be said to be more democratic inasmuch as they thought that people's rights would be best protected if they had direct influence over them.
https://teachingamericanhistory.org/resources/fed-antifed/antifederalist/