The public funding for primaries was conditioned of following certain rules in the Democrat party charter? Cite?
I don't believe it needs to be outright explicit. If, however, they get taxpayer dollars in order to run a primary, they are essentially saying that they are going to use the money to run X event using Y rules. If they don't do that in exchange for getting the money, they have essentially defrauded the taxpayers.
In any case, the public should not be funding the primaries. The problem there being that the public is largely represented by the Democrats and Republicans.
I think it's fine to have the voting for political party nominees to be funded by the pubic. It's using the same organizational structure that's already there for the general election and it would be a waste to have parties duplicate that process. When they do so, however, they can't then claim to be nothing but private organizations who are dealing with purely internal matters.