For move 13 I don’t know when to play R14, R17, O17, or P14. Is it something humans can understand? Or as far as humans are concerned are they all just as good?
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not sure what you mean by playing R14 since that’s been played there
But aside from that whole nitpick, the question there is one of strategic nature rather than tactical. They’re all good moves, but what you really wanna ask is what’s important to this board state?
R17, as you played, is the territorial option. It says “welp, corner’s open, thank you!”, and simply looks to take territory. It’s often the AI’s preference, but it gives black the choice on which way they want to block, and also the influence on that side that they choose. Hard to go wrong with this one
O17 is a little stranger, it says “well, the corner is nice and all, but I’d really like to have a position on the upper side, with a bit of aji as you smother the first stone I played.” You don’t see it as often, because that stone at R14 is a big thing to practically sacrifice, but in the right strategic situation can be a very useful option.
P14 is a lot more bold, it says “territory? In this position? I’m just going to keep you separated” and asks for either a fight, or to lean on Q12 tor influence, without really sacrificing prospects in the corner. It’s often recommended that you have strength on the lower right, or somewhere in the vicinity to aid in the potential ensuing fight, if you choose this move.
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Thank you very much. This issue has been on my mind for a while.
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I’d want to reword it to a more brief and formal framework to put it into an encyclopedia,
but if you feel my words would be useful for it, I’m not against them being used there
As far as most amateurs are concerned, they’re all just as good 
Personally I would have played the same move as you in this case. KataGo tells me I’m wrong, but only by a fraction of a point. We amateurs are going to make much bigger mistakes later in the middlegame! So just try to play reasonable moves, experiment, and see what suits your style. All of the options that you listed are reasonable.
More important is getting the followup moves right. White made a couple of mistakes and you got a good result from this corner. But then S16 doesn’t help you at all: you should go straight to S18 on move 23. By sacrificing S16, you make white’s territory more solid, which costs you a couple of points later.
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