Someone with perfect strategy would have an answer for any deviation. Playing imperfectly would likely get you into a losing position. There is no way to go from a game being drawn if played perfectly to being winning if the then loser were to be using perfect strategy.
Imagine finding a pamphlet written by God that contains a listing of a perfect game of chess. Armed with it, you will still lose easily to Magnus Carlsen. He will deviate fro| the sequence and you won't know the perfect responses to his moves.
How would you go to prove that this is the perfect game of chess, if you didn't explore all of it's possible sequences? If they did solve the game, there's no way around knowing all possible outcomes - starting from a fixed given position.
An "ultra-weak solution" is even less. It gives the win/lose/draw outcome of the perfect strategy, without producing that strategy, nor even producing the game of perfect play from the initial state.
This is all covered in the second paragraph of the paper's introduction.
https://en.wikipedia.org/wiki/Solved_game
And if a strategy for perfect play only from the initial position is a strong solution, then what do you call the even more strongly solved case of a strategy for perfect play from any position?
The second paragraph of the paper isn't super specific about its definition for "weakly solved", but I read it as agreeing with my statement. It also calls checkers "weakly solved", a game for which a strategy to beat any possible move is known.
Even if you don't know the moves you will know that you are in a winning / drawn position. Magnus can not find a way to get to a winning position unless you make a mistake.
Perfect play in a perfect game means that the objective won't change - in optimal control theory this is called Pontryagin's Maximum Principle or the Bellman optimality criteria. A "value to go" function is enough to find the optimal solution.
That's the easiest part.
Someone verified there is a perfect strategy, using tons of computational resources and lots of time. The game tree they explored may have millions of billions of nodes, of which only the top layers could be saved.
To use a perfect strategy in an actual game, you need to have it stored in some format that allows near-real time lookups.
Such is the difference between weakly and strongly solving.
EDIT: Strongly solving goes beyond the latter, requiring real-time best play from arbitrary positions.
no, you described difference between storing the result and not storing it
strongly solving means allowing arbitrary amount of mistakes from either player before giving to the solver
This does have to be mutual though, if only one side plays imperfectly, then the other can always force at least a draw.
So if you know the other side is incapable of playing perfectly, it could be rational to deviate.