It's like tonic, which isn't considered a mixed drink ($0$ parts gin and $1$ part tonic). That's degenerate for you: "In mathematics [as in mixology], a degenerate case is a limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class."
If you like, you can think of a pure strategy as a mixed strategy in which a player has a 100% chance of picking a certain strategy.
The equilibrium definition is the same for both pure and mixed strategy equilibria ("even after announcing your strategy openly, your opponents can make any choice without affecting their expected gains"). The difference is that in a mixed equilibrium, you are announcing your probability distribution, not the strategy that it randomly produces.
Example: Rock-Paper-Scissors. There are no pure strategy equilibria: If I announce "I'm going to play definitely Rock!" then clearly my opponent will choose Paper; if I know they're going to play paper then I don't want to play Rock anymore, so this is not stable. However, if I announce "I'm going to secretly roll a die, play Rock if it shows 1-2, Scissors for 3-4, and Paper for 5-6!" then my opponent is equally happy with any choice he makes. If he therefore chooses the same strategy as me, then I am equally happy with any choice I make, so this is a mixed equilibrium.
Also, your statement "the linear equations of the mixed one can only give one or infinite number of results" isn't true - there are many games with an in-between number of equilibria (Chicken, for example, has 3 in a two-player version of the game). As an aside, a randomly-generated game will have a finite, odd number of equilibria with probability 1, but that's about all you can say about the number of equilibria.
Best Answer
The set of strategies available to a player is a full list of the choices he can make.
There are two major conventions to interpret this definition. The first one, following natural language, is that there is a set of choices that the player can make. These "native" options are called pure strategies. F.i., in a Prisoners' dilemma, you can stay mum (cooperate) or incriminate your partner (defect).
A more extensive interpretation is that a player can inject randomness in his choice, by selecting a randomising device to pick one of his pure strategies. F.i., he may decide to toss a coin and then cooperate/defect based on getting heads/tails. The options to randomise over pure strategies are called mixed strategies. The set of pure strategies can be associated with the subset of mixed strategies that pick a specific pure strategy with probability one.
The mathematical motivation for mixed strategies is that they are necessary to prove existence of a value in zero-sum two-player finite games or existence of equilibrium in finite games. An intuitive motivation is that sometimes your best option is making yourself unpredictable: f.i., think of playing hide-and-seek --- if you choose your hiding spot randomly, you give the seeker no clue where to look for.