What doesn't carry over (becomes a normalized amount)
- Base defense
- Base weapon attack
What does carry over
- Weapon progression (including damage bonuses)
- Armor progression (Including Defense bonuses)
- Abilities
- Character progression (stat upgrades)
This means that even though they are on relatively equal starting ground. A Level 20+ will have an advantage over a level 5, but not so much that the low level doesn't stand a chance.
To directly answer your question instead of quoting long winded and confusing reddit posts:
No, upping your light level will not make you more effective in terms of damage on content unless you were previously underlevelled for it, assuming the same weaponry. Your weapon reaches its maximum damage the moment you hit the same level as the enemy upon which you are firing (unless you're using severely underlevelled weapons, see below), and it will stay that way against all enemies you are of equal or higher level to. The only way to increase your DPS is by upping your weapon's attack or getting higher attack weaponry (or I guess unlocking certain armor mods, but that's besides the point).
Sidenote: This does NOT apply to abilities, just weaponry. Your abilities increase in damage as your light amount goes up.
You will, however, benefit from the better abilities that are on those pieces of equipment and the increased armor rating, which will allow you to survive better. You will also presumably have faster recharging abilities due to the higher level of stats on better gear.
The only way level applies to damage, then, is down scaling due to being underlevelled. If your level is below the level of the enemy you are fighting, you take sharp damage penalties, at four levels below you are not damaging the enemy at all (its level will show ??). There's also a damage penalty if your weapon's "internal level" is much higher than the enemy you are fighting, but that's rarely very useful information. It does, however, mean that you won't suddenly do worse against low level content just because YOU levelled up, if you're still using the same gun.
Best Answer
Yes, you are correct. In order to advance in the campaign, you must also advance in levels. To level up one may play-through the campaign missions to receive xp, play in the crucible, complete bounties and strike missions as well as other ways. Unfortunately this game is less of a rent for the weekend and more of a spend weeks grinding for weapons and armour in raids and weeklies, so if that is not your desired play style than this is probably not the game for you.