"[Input] parameter" and "[non]deterministic" are not at all unique to MySQL.
Generally speaking, a function or routine is deterministic if it always produces the same output for the same input. The classic example of a routine that isn't deterministic is one that depends on one or more time and date functions (e.g. NOW()), or on the RAND() function.
As for when to use the DETERMINISTIC keyword, I think the Manual is pretty clear: If you don't use it, the optimiser does not attempt to perform some optimisations that it would otherwise; if you do use it, be sure that your routine is in fact deterministic, because using DETERMINISTIC for a nondeterministic routine causes the optimiser to make bad choices that can have a negative impact on performance.
Your routine does not actually return a value, but it does perform an update; in this case, you must consider the values inserted, one of which depends on NOW(). And the Manual is quite clear that "[a] routine that contains the NOW() function (or its synonyms) or RAND() is nondeterministic," and so contains the answer to that question.
MySQL Documentation Team @ Oracle
Orlando, Florida, USA
MySQL Dev Zone
MySQL Server Documentation