Resistance and advantage are two separate mechanics

Advantage has to do with rolling the dice, resistance has to do with reducing damage in half.

Easy illustrative example is a Dwarf

Dwarven Resilience. You have advantage on saving throws against poison, and you have resistance against poison damage (explained in chapter 9). {Basic Rules, p. 15}

The dwarf rolls 2d20 whenever she has to make a saving throw against poison. Because a dwarf is resistant to poison, any poison damage she takes is halved.

Resistance

If a creature or an object has resistance to a damage type, damage of that type is halved against it. {Chapter 9}

Advantage and Disadvantage

... ability check, attack roll, or saving throw is modified by special situations called advantage and disadvantage. ... When you have either advantage or disadvantage, you roll a second d20 when you make the roll. Use the higher of the two rolls if you have advantage, and use the lower roll if you have disadvantage. {Basic Rules, p. 5}

What to share with your DM from the rules

A creature with advantage on saving throws versus magic/spells still rolls the saving throw: 2d20, pick the higher score. Resistance is applied separately once the amount of damage is determined by the saving throw's success or failure.

Resistance and then vulnerability are applied after all other modifiers to damage.
{Basic Rules, p. 78}

Dwarf Example:

Exceka the dwarf is hit by a giant scorpion's tail; the damage roll for poison is 24. Exceka has advantage on the saving throw. She rolls a 3 and a 19, she saves. Damage is halved, so she takes 12 damage, except, she has resistance to poison! She takes half of that damage: 6 HP is the final damage.
_{FWIW, the wizard abjuration school has a feature that grants both resistance to spell damage and advantage on spell saves. If one of them implied the other, it would be redundant and unnecessary to specify both. Thank you, @Ryan C. Thompson}

Demon Example: Balor

What Magic Resistance means for this demon is explicitly spelled out in its stat block.

Magic Resistance: The balor has advantage on saving throws against spells and other magical effects. (MM, p. 55)

That is different from the Damage Resistances in the stat block (MM p. 55) that says

Damage Resistances Cold, lightning, bludgeoning, piercing, and slashing from non magical weapons

And that is different from

Damage Immunities Fire, Poison

And that is different from

Condition Immunities Poisoned

To illustrate: Balor is hit by Ice Storm spell (cold + bludgeoning damage)

Each creature in the cylind⁠er must make a Dexterity saving throw. A creature takes 2d8 bludgeoning damage and 4d6 cold damage on a failed save, or half as much damage on a successful one.

A Sorcerer cast the spell, and rolled damage: 12 bludgeoning and 16 cold.

1. Step 0. Is the spell a magical effect? Yes. Apply advantage to the saving throw.

2. Step 1. Is there a saving throw? Let's say the Balor rolls a 9 and a 17; spell caster's DC is 16. Balor saves.

3. Step 2: Save -> half damage.
   a. The magical bludgeoning damage becomes 6. Demon is not resistant to magical bludgeoning attacks. (The barbarian with the non-magical maul who did 16 damage last time only did 8, due to Damage Resistance cited above)

b. The cold damage becomes 8.

1. Step 3: Now apply resistance.

Balor is resistant to cold, so it takes only 4 points of the 8 cold damage.

Final note

Damage is by type. Magical is not a damage type (but it is a kind of attack or effect). The damage types are on basic rules p. 78, and in PHB p. 196:

Acid, Bludgeoning, Cold, Fire, Lightning, Necrotic, Piercing, Poison, Psychic, Radiant, Slashing, Thunder