DND_Reborn
The High Aldwin
Thanks for all the feedback! Awesome stuff. 
I'm not worried about devaluing the monk's feature. At best, a cleric with investiture and the Resilient feat could have five saves, which would be very strong but only at higher levels with Resilient. Most other classes will get 3 to 4.
I wouldn't be opposed to 1.5x proficiency if someone applied the 1/2 prof save gained later on to one of their initial save proficiencies. If you really wanted to be that good, then that would be up to the player. Personally, I play more diversified myself, but others like to be super-focused.
I forgot to include swim with Move in the Wild. I'll add it later.
Revealing Sight mirrors more the feel of the AD&D Unearthed Arcana barbarian. I like the concept of barbarians shunning magic more than embracing it. That's my preference so I tried to tie it back to the old barbarian concept.
Something like this would work, sure, or might make a nice feature for the Ranger class. I'll add it to my notes and see if I want to use it.
They don't "channel" magic. Think of it more as an "anti-magical aura" that allows the barbarian to harm creatures which require magical weapons. Like their Magic Resistance feature, if they associate too much with magic, their powers suffer sort of. Again, this is just following the old AD&D barbarian.
The Powerful Strike, let's call it, might work well as a bonus action?
Powerful Strike
Your rage allows you to use your might to topple your foes. Beginning at 6th level, if you successful hit a creature with a melee weapon attack while raging, you can use your bonus action to shove the creature 5 feet away from you or knock it prone. The target must make a Strength saving throw or Dexterity saving throw (its choice) with a DC equal to 8 + your proficiency bonus + your Strength modifier.
Durable Instincts
Your rage blocks off your mind from outside influence, making attempts to influence or control you difficult. While raging, you have advantage on saving throws against spells and effects that charm or dominate you. In addition, Persuasion and Intimidation checks made against you while you are raging have disadvantage.
Pretty close to your ideas. What do you think?

Ok, first set of features:
I like the aspect of choosing saving throws to gain half-proficiency in. I think the only concern I have with this is that it devalues the monk's feature to gain proficiency in all saving throws, and characters who really want a particular saving throw that their class doesn't provide (e.g., casters with CON) are going to be torn between taking Resilient CON early and then losing out on the big benefit of this feature, or waiting and having a rough time with that save before then. What would you think about allowing this half proficiency to stack with proficiency from another source (up to a max of 1.5x proficiency)? It sounds strong, but it's a sizeable investment to get there, and it's still weaker than the Paladin's bonus to all saves
I'm not worried about devaluing the monk's feature. At best, a cleric with investiture and the Resilient feat could have five saves, which would be very strong but only at higher levels with Resilient. Most other classes will get 3 to 4.
I wouldn't be opposed to 1.5x proficiency if someone applied the 1/2 prof save gained later on to one of their initial save proficiencies. If you really wanted to be that good, then that would be up to the player. Personally, I play more diversified myself, but others like to be super-focused.
Barbarian:
Unfaltering Stamina: Love it. Very on-theme; not overly powerful, but certainly useful. Makes berserker a little more viable.
Move in the Wild: Nice. Removes the temptation to spend a rage just to get advantage on high stakes climbs or jumps.
Revealing Sight: A cool ability, but I'm not sure what it has to do with being a barbarian. It's sort of a higher level, weaker version of the warlock invocation Eldritch Sight. Something should go in this level since Brutal Critical doesn't seem like enough for a whole level... but I'm not sure this is it.
I forgot to include swim with Move in the Wild. I'll add it later.
Revealing Sight mirrors more the feel of the AD&D Unearthed Arcana barbarian. I like the concept of barbarians shunning magic more than embracing it. That's my preference so I tried to tie it back to the old barbarian concept.
What about something like
"Primal Intuition: You have an instinctive ability to sense that something is off when your senses conflict. You gain proficiency in Insight if you do not have it already. If you are already proficient, you add one and a half times your proficiency bonus (round down) to all Wisdom (Insight) checks. Whenever you make a Wisdom (Insight) check to assess whether another creature is hiding hostile intentions, you can treat any roll of 9 or lower as a 10. In addition, when a illusion that affects a living creature would normally require an Intelligence (Investigation) check to detect its presence, you can choose to make a Wisdom (Insight) check instead."
Something like this would work, sure, or might make a nice feature for the Ranger class. I'll add it to my notes and see if I want to use it.
Magic Weapons + Magic Resistance: I see why you put this here given the next feature, since you're trying to push barbarians away from using actual magic weapons, but... it's not clear why barbarians would suddenly start channeling magic through all their weapon attacks. I think this ends up doing the opposite of what you were going for, in trying to keep barbarians a "non-magical" class.
They don't "channel" magic. Think of it more as an "anti-magical aura" that allows the barbarian to harm creatures which require magical weapons. Like their Magic Resistance feature, if they associate too much with magic, their powers suffer sort of. Again, this is just following the old AD&D barbarian.
For 6th level if you are looking for an offensive boost, how about adding some sort of shove rider to your weapon strikes? "You can put extra force behind your weapon attacks. Once per turn when you hit another creature with a melee weapon attack, you can make a Strength (Athletics) check, contested by the creatures Strength (Athletics) or Dexterity (Acrobatics) (its choice). If you win the contest you can shove the creature 5' away from you or knock it prone." You can of course do this normally by spending an attack; but this lets you do it after you hit with a weapon.
For 8th level how about something like: "Durable Instincts: While raging, the link between your natural instincts and you behavior becomes more difficult to interrupt. You have advantage on saving throws against Intelligence, Charisma and Wisdom saving throws to resist effects that would compel you to act contrary to your normal behavior." (This steps a little bit on the berserker's charm+fright immunity though, so might need tweaking).
The Powerful Strike, let's call it, might work well as a bonus action?
Powerful Strike
Your rage allows you to use your might to topple your foes. Beginning at 6th level, if you successful hit a creature with a melee weapon attack while raging, you can use your bonus action to shove the creature 5 feet away from you or knock it prone. The target must make a Strength saving throw or Dexterity saving throw (its choice) with a DC equal to 8 + your proficiency bonus + your Strength modifier.
Durable Instincts
Your rage blocks off your mind from outside influence, making attempts to influence or control you difficult. While raging, you have advantage on saving throws against spells and effects that charm or dominate you. In addition, Persuasion and Intimidation checks made against you while you are raging have disadvantage.
Pretty close to your ideas. What do you think?