Much much fail in your thoughts ... first of all we have to consider all game elemets if we want to categorize in genres b/c u see a classic Adventure game is nothing without it's story rly ... no one cares about the ill logic puzzels it's all about the story. Now no one can blame you for beeing confused about the genres. The main "problem" is that for generating a stronger feeling of progress you have the RPG elements in nearly every game so it's realy hard to consider which game is an RPG an which not. On a side note the genre Action-Adventure is crapiest bull crap of crap crap b/c if someone say to me this game is an Action-Adventure is generealy have no idea what this game lookes like. Just compare Metroid Prime to Super Metroid or Ico .... now there are similarities but they are also shared by other genres.
Now from what i know there are some core elements which define genres and allthough i can't remember all of them here are some which could be interesting:
Adventure: Story
Shooter: Level Design
RPG:Character Progression
Jump'n'Run: Player Progression
RTS: Level Design
These are the primary things which drive me to play on in the corresponding genre. So for the debate: RPG vs Strategy. You have to admidt that there are some elements of strategy in the RPG genre like water is good vs fire etc but are these things what drive you through the game? Are realy these little strategic moments the force behind the game? Or are you interested in maxing your character out, exploring his story or creating your own story with this character to make him the brave warrior or the bright wizzard he could be? Therefore you have mostly different paths you can go in true RPGs. For example the spheroboard in FF10 or the different classes in Secret of Mana 3. In game which only get some RPG elements for the better overall feeling there is simply just one possible way or the cusomization options are not very broad like for example in Batman:AA (you eventually just get all the stuff) or Infamous.
Now strategy games generaly involve very little of overall advancement. There are tiers, but progress is not permament and in the next level you have to start from the beginning. The most important thing in strategy games are the levels, that's partly why it's so important to have fog of war. The player have to explore the map, the player have to be surprised and awarded for exploration. Like killing creep camps in WC3 to get items or exploring a hidden mineral field in C&C:RA. Now ofc the battle is big part of RTS but most of the time it's building base adopting to the Level Design and to new units(yours and of your enemy) and gaining an edge in resources.