To really differenciate the races they should have skills that no other race can get even through spells. Nords for example could be 100% resistent to ice effects whereas the highest enchantment item you could get for it would be only 75%.
This is only an example I'm sure others will come up with better examples of what could be done.
I think wood elves for example should rather than just being short (which is the only way to identify them in Oblivion), should have ridges along the nose or across the forehead or something different in their apperance than just being short.
Race specific perks are also a good idea
I am against your first example. Natural resistances (Such as Nord Frost Resistance, Breton Magic Resistance, and Dunmer Fire Resistance) are good enough to not have to be maximized: They free up an enchantment slot, so you have a
much easier time gaining 100% Immunity to your natural resistances.
The mods that made Khajiit Eye-of-Night and Argonian Permanent Water Breathing are a good example of how those two races could have always notable differentiation.
If the tech's out there, I hope they can make Khajiit darkvision dynamic and unique from Nightvision spells, so it automatically adjusts to the right brightness needed to function, possibly by increasing the Contrast of the brightest part to be amplified to Almost White, and have the in-between sections correspondingly scale. A functionality no spell could replicate, but it's not as obviously broken as your suggestion of One-of-a-kind Frost Immunity for Nords. (Theives might disagree, because they can't hide from Khajiit!).
With stats gone, they could further differentiate the races with percentile-based bonuses:
Redguards would be the most "Proficient" warriors have greater effectiveness from their block and weapon skills, such as 10% greater effect than their skill level would normally indicate, or the ability to ignore a small % of the damage reduction from armor. They could also gain slight move and attack speed bonuses the more injured they are, from Adrenaline Rush. Thus, they get greater emphasis on "Weapon Mastery" fighters.
Nords would be the "Strongest" warriors. getting their Frost Resistance, as well as a natural, static-percent Damage reduction to give them melee survivability, but most of the bonus would be greater Impact (Stagger chance) and wider area of attack with two-handed weapons.
Orcs would be the "toughest", getting a flat percentage bonus to damage (Outpaced by Redguard damage bonus at higher levels, but superior at lower ones), as well as treating armor worn as having +10-25% greater base protection value. They would also have a resistance to magicka.
Khajiit could be more mobile in combat, but less stable (As I suggested earlier) In addition, they also jump higher, further, and have greater mid-air control of their jumps than other races, on top of their nightvision.. Perhaps greatly increasign their "walk" speed in combat would achieve the first goal of greater mobility. A 10% bonus to Fire Damage inflicted (Fuzzy pyromaniacs) would also be appreciated, even if it comes with a vulnerability to fire as well.
Bosmer should be harder to detect because of their small profile, and have a bonus to draw speed with a bow. The ability to calm
any animal (As
a lesser power) or control a number of them (Greater Power) are also some ideas.
Imperials should get a bonus to the effects from the Dragon Shouts, discounts for selling and buying, the ability to fence any stolen goods to anyone, the ability to talk themselves out of certain fines, and a lower chance of being "recognized" by intelligent enemies (Making them the third "Thief" race)
Bretons have their magic resistance, and could have 50% greater Base Magic than other races. This more than outpaces the slight Argonian spellcasting bonus.
Dunmer get their fire resistance, a small magicka bonus, and reduced mana cost for destruction and conjuration spells because of their Daedric connections.
Altmer have a weakness to Magic that applies to both negative and beneficial effects, as well as double the base magic of non-spellcasters.
Argonians should have a 50% reduction on the effect Running has on Stamina, making them better athletes. They can't drown, and also are immune to poison, and strongly resistant to disease. Their skin coloration should give them a slight advantage in avoiding detection. and they should have a small bonus to the cost reduction of spells from skill. A small natural armor bonus might also fit, making them a nice "Jack of All Trades" race.