What is a precise definition of "in-place"?
An in-place algorithm assumes its input is given in working memory, and it cannot use any other working memory than what the input initially occupies. We can consider any fixed amount of working memory to be part of the algorithm itself.

For example,


finite automata are in-place;
pushdown automata are not;
linear bounded automata are;
Turing machines are not.


A formalization of the notion can be a random-access machine with limited memory.