First off, a definition of a "normal" primary for the purpose of the this thread: Candidates file with a government agency, appear on a ballot amongst members of their own party, and then the winner of each primary goes on to the general election. Run-offs are allowed if no one reaches 50%
Which states differ from this, and in what way? A few I could think of off hand (although my details may not be complete or accurate):
Washington: Top-two primary
Minnesota: Seems to rely heavily on conventions to determine nominees
Virginia: More conventions
Texas: Administered by parties and can't occur in a county without a county chairman
South Carolina: Also run by parties
Any more details or any other states?
An important distinction is who may vote in primary elections.
There are:
Open primaries: Voters may choose a primary on election day.
Closed primaries: Only voters registered with a party may vote in its primary.
Semi-closed: Only voters registered with a party, plus unaffiliated voters may vote in its primary.
The Supreme Court has ruled that a State party may have a semi-closed primary even if this contradicts State law and the State is conducting the primary. That is, a party may open its primary to non-affiliated voters. But the Supreme Court has also ruled that a party may
not open their primary to registrants of other parties.
In some States, unaffiliated voters may register with a party on election day, blurring the distinction between semi-closed and closed. In some States, voters may also switch party registration on election day.
Some States do not have party registration which means their primaries are open. In some of these States, voters choose their party in secret and there is no public record of which party's primary a voter voted in. In other States, a voter chooses his party, and there is a public record of the activity, but this is not binding on elections in future years.
States also have different procedures for being placed on the primary ballot, on how minor parties are handled, and how independent candidates are handled.
While it is true that in Texas, primaries are administered by the parties, they do so according to State law, so that the difference is generally superficial from both the perspective of voters and candidates. It is worthy of a footnote, not a classification distinction.