As someone with a little bit more of a math background, I call these assumptions constraints. The word comes from optimization, a field studied by both computer scientists and mathematicians. When you write a computer program to solve a problem like determining the most efficient way to schedule trains transporting materials, you provide the program constraints that limit what kinds of solutions are acceptable. For example, a weight limit on a train is a constraint, as is the fact that two trains can't go in opposite directions on the same tracks simultaneously.
Analogously, a basic assumption or constraint limits what solutions are possible in a puzzle.
Another word you might want to use to refer to the basic assumptions is givens. Again, this word is taken from math: When completing a mathematical proof, the ideas or objects that you're making a proof about are given to you at the start of the proof. For example, let's say you want to prove the Pythagorean Theorem: If a triangle is a right triangle with legs of length $a$ and $b$ and hypotenuse of length $c$, then $a^2+b^2=c^2$. The givens in this proof are that the triangle you are working with is a right triangle, and it has sides of length $a$, $b$, and $c$. If you take these details out, you don't have enough information to complete the proof!
Likewise, the assumptions you're allowed to make while solving a puzzle are the puzzle's givens. For example, in your problem about knights, knaves, and jokers, one of the givens is that the three characters know who each other are.
I don't think there's too much of a difference between a constraint and a given when discussing puzzles. If you want to be strict about it, though, a constraint is an aspect of a puzzle that puts requirements on an acceptable solution without decreasing the size of the search space (a hypothetical list of every possible solution based on a naïve reading of the puzzle), while a given is either necessary to ensure the puzzle can be solved or helps decrease the size of the search space. In other words, a constraint makes the puzzle harder by forcing you to find more complicated solutions, while a given makes a puzzle easier by pushing you in the right direction or ensuring a puzzle has only one provable answer. One can see where the difference can get blurry for a lot of puzzles.
