[Economics] On the (rather strange) notion of strategies in repeated games
Your proposed notion of what a strategy should be in repeated game can be adapted to all extensive form games has been called a plan of action by Ariel Rubinstein, who discusses this and related issues in the paper
Rubinstein, Ariel. "Comments on the interpretation of game theory." Econometrica: Journal of the Econometric Society (1991): 909-924.
Some concepts become harder to define, such as backward induction in games of perfect information. However, one can still do pretty much all of game theory. The book " Rationality in extensive form games" by Andres Perea uses the definition as a plan of action throughout.
The standard definition of a strategy assigns an action to every decision node/information set. This definiton is at least a jack of all trades; it works well in most cases and sometimes leads to slightly annoying notations that seem overcomplicated. I argue that changing the definition of strategy on a case-by-case basis would probably lead to more confusion. (You can refute me by proposing an exact yet general definition that accomodates your concept.)
Also note that if you do not assign an action to every decision node/information set, in a more complicated game you may run into trouble when you examine different strategies of the other players, and suddenly you reach a decision node that you mistakenly thought was unreachable. (This is also a common mistake made by undergrads.) Assigning an action everywhere is a safe bet.
Finally you would also need to adjust other definitions, such as subgame perfectness, to fit your new concept of strategy, as it would leave subgames without any actions assigned.
I don't find the definition of a strategy strange at all. In contrast, it seems to be the most natural way and other ways seem to run into problem once applied to general games.
As Giskard said, a strategy maps a decision node/information set into an action. In repeated games, this corresponds to mapping a history (what has happened so far) into an action. To evaluate the profitability of a deviation it is necessary to specify off-path behavior.
Draw a game tree like this: Player 1 can play A and payoff is (2,0), or play B after which player 2 can choose a or b. After B-a payoff is (0,2), after B-b, player 1 can choose C or D with payoffs (5,2) and (0,0). Now would you say "player 1 plays A" is a full strategy? After all, the game is over and all off-path behavior is irrelevant. Of course this does not make sense, because 1 needs to know what 2 does in case 1 plays B. Otherwise who knows if a deviation to B pays off? Maybe 1 would get 5 instead of just 2 - or she gets 0. So let's specify that 2 plays a. Then 1 knows that deviating 2 B does not pay off.... but what about 2? Now, 2 doesn't know what would happen if she played b. So we need to specify that, too, and so on.
Your example of a repeated one-player game is weird to begin with. Why should there be a repeated game of just one player? It is just a repetition of the same game and past behavior does not influence future play. But even in this game a strategy should be a full plan of actions. Suppose you have made a mistake and played B in the first period, then what do you do in the second period? The strategy you propose can also be captured verbally by "Always play the action played in the previous period. Play A in the first period." Does this sound strange to you? In repeated games, you usually do not write down an action for every single history one-by-one, because the main interest is on infinitely repeated games and there are infinitely many histories.
Let's take an infinitely repeated prisoner's dilemma. How would you specify a strategy? If you only specify that both players cooperate(C) if both cooperated before, you cannot evaluate if a deviation to defect (D) is profitable. How would your "more natural approach" take care of this? What would be the notion of subgame perfection? For instance, the grim-trigger strategy "Play C in beginning, play C if opponent always played C so far, otherwise play D forever" is a Nash equilibrium for high discount factors $\delta$. However, it is not a subgame-perfect Nash equilibrium for any $\delta$. In contrast, the grim-trigger strategy "Play C in beginning, play C if everyone always played C so far, otherwise play D forever" would be subgame-perfect for large $\delta$.