Again where is the confusion... You seem to want to make it confusing when it's pretty clear when you use which... No skill system can be a detailed list of everything you can do... but you do want to cover the bases in your particular genre and I do think in the sources D&D pulls on genre wise... the ability to play an instrument without being a full on Bard is present enough that it should be accounted for and in 5e it is...YMMV of course. Just curious what skill do I use in 4e if I want to play a musical instrument but am not a Bard?
Why would a check ever be required to play a musical instrument? If you want to DO something, then check on the success of the thing you're doing. Are you influencing someone, Diplomacy, are you inspiring them, CHA, are you scaring them, Intimidate, etc. There's no use at all for any numbers to be attached to playing an instrument, and its not a character resource if it isn't something that matters. In 4e you can just say "yeah, I play my fiddle to raise everyone's spirits", but you could as easily talk to them etc and get the same results. A player is free in 4e to simply write on his sheet "background, knows how to play the fiddle" PHB2 even provides a bit more rigorous framework for that if you want to use it.
Using Investigation is an Intelligence check... One can definitely be trained in logical deduction as a skill set (there are mental exercises people perform to do just this), it is basically the "Sherlock Holmes" skill. I think you're inability to use it was probably based more on your DM not structuring situations where it was useful... which can happen to any skill if the DM does not account for them... or your lack of understanding in when to apply the skill...
Sherlock Holmes is super intelligent and has preternatural perceptual abilities. So he has a high INT and a high Perception bonus. He has also studied a wide variety of things, which in 4e terms might be represented by training in some or all of the knowledge skills.
Claiming that one can't be trained to better use deductive reasoning is like claiming that one cannot be trained to better utilize strength in athletic activities...
But I can't find a distinction between INT as an ability score and deductive reasoning capability. This is the same issue that 4e has with Endurance as a skill, to be 'trained in endurance' is simply to have a higher Constitution! Someone who has inured himself to environmental conditions, etc surely would simply have more hit points wouldn't they? There were already feats like 'Toughness' which captured the possibility of additional benefits, but there's just no distinct thing to 'learn' to be tougher. Likewise 'learning deduction' is IMHO just an INT increase. It should apply across the board and automatically to any endeavor based on intellectual ability.
I would assume that if someone is not using something properly... it would be harder thus a penalty... not that said thing that allows you to use a skill properly in fact gives you a bonus on top of your skills...
Meh, they could have written it either way, but 4e was based on a 'give bonuses not penalties' formula, so its a bonus. I don't think it really matters.
Again, if this is clear what is the confusion around 5e... the mechanical application of thieve's tools in 5e is just as clearly stated...
But I cannot have any proficiency with locks, traps, etc independent of this set of tools? It seems like a skill to me, and I just don't understand why there are 2 categories. This is also true with some of the other 'tool' proficiencies, they seem to imply knowledge and other things which are separable from the materials that the tool proficiency is bound to. Its an awkward design.
I am literally looking at my Rules Compendium right now at the Dungeoneering skill and it makes no mention of psionics or being applicable to that area... to start off can you quote where that information is? I also don't see anything around Primal magic in the write-up of the nature skill...could you also quote where that is stated?
Note: I am also still wondering what shadow magic falls under?
Dungeoneering covers everything related to the Far Realm and Aberrations. Psionics falls under that rubrick. This is a bad example though because Dungeoneering is presented in PHB1, and psionics don't appear until PHB3, and aren't in Essentials at all (though admittedly there is an explanation of psionic points IIRC in the RC). PHB1/RC DO tell us what sorts of monsters are covered by Dungeoneering however, and the game clearly associates psionics and Aberrations, so its not actually a HARD question to answer, though one of the most obscure ones on the skill front.
Nature covers all 'Natural things' and natural creatures/animals, which being part of nature of course fall under the Primal power source since primal power is the power OF nature. The question is then only one of 'what is magic'? and 4e doesn't really clearly define things as 'magical' or 'not magical'. In fact the Arcana skill is a bit of an anomaly in terms of actually discussing 'magic' as a specific category of thing, the rules do so nowhere else, except perhaps in defining items as being 'magical'.
Shadow Magic falls under Religion, which covers all things and phenomena associated with the Shadowfell, which is the source of shadow magic. Again, 4e doesn't try to pin down what is or isn't 'magic'.
In general 4e doesn't try to define things that have ordinary definitions and aren't mechanical elements. Magic is a narrative concept, with its ordinary accepted meaning. Anything which is not mundane, able to exist in the real world, is perforce 'magical' in nature (and there is a keyword 'magical' for such creatures). As a result Arcana can detect the presence of MANY things in 4e, including things covered AS KNOWLEDGE by other skills. In other words you wouldn't detect 'Primal Magic' using Nature, you'd detect it using Arcana because 'Primal Magic' implies something supramundane. The Nature skill MIGHT also detect a Primal effect ("that's not right, polar bears never live this far south!") and it could give you knowledge about the effect ("The polar bear is the sign of the Spirit of Winter's son, the Lord of Ice!").
In some ways it would have been simpler if 4e had precisely defined 'magic' as a mechanical concept in 4e, but that would have had some big downsides as well, like by implication making the fighter 'mundane' or 'magical' in an explicit way.