Is there a name for the do -> recurse -> undo pattern in backtracking?

In backtracking, i.e. the algorithm used for solving the n-queens problem, there are basically two ways to do the recursive call:

1. copy the parent board to make a child board, modify the child board by placing a new queen, then do the recursive call on the child board.
2. modify the board directly, do the recursive call, then undo the modification.

The second is preferred since it avoids the costly copy.

This choice is also present in other algorithms, like minimax on games.

Is there a name for pattern 2 as opposed to pattern 1?

In Constraint Programming and SAT-Solving (where your n-queens example usually comes from) i would argue, that these concepts are described as:

• copy-based
• trail(ing)-based

See for example:

Reischuk, Raphael M., et al. "Maintaining state in propagation solvers." International Conference on Principles and Practice of Constraint Programming. Springer, Berlin, Heidelberg, 2009.

Schulte, Christian. "Comparing Trailing and Copying for Constraint Programming." ICLP. Vol. 99. 1999.

Excerpt of the former:

Constraint propagation solvers interleave propagation, removing impossible values from variable domains, with search. The solver state is modified during propagation. But search requires the solver to return to a previous state. Hence a propagation solver must determine how to maintain state during propagation and forward and backward search. [...] This paper also provides the first realistic comparison of trailing versus copying for state restoration.

Both have their pros and cons, analyzed in the references.

Keep in mind, that the trail is usually not only about storing your decisions (board placements), but also propagations happened (this placement leads to these now impossible placements due to alldifferent propagation: these effects must be reverted as well!). For an overview of the implementation of such, see: MiniCP