Damage Threshold in the D&D 5e SRD is described as:

"Big objects such as castle walls often have extra resilience represented by a damage threshold. An object with a damage threshold has immunity to all damage unless it takes an amount of damage from a single attack or effect equal to or greater than its damage threshold, in which case it takes damage as normal."

For example: A warship with a Damage Threshold of 20 takes 50 hit points of damage from an attack. Does the warship take the full 50 hit points of damage? Or does the warship take 30 hit points of damage after adjusting for the Threshold?

The warship takes all 50 damage

In the quote you provided, it says (DMG, pg. 246):

An object with a damage threshold has immunity to all damage unless it takes an amount of damage from a single attack or effect equal to or greater than its damage threshold, in which case it takes damage as normal.

"It takes damage as normal" means that it would take the 50 damage as though it didn't have the threshold. The only difference here between objects with and without thresholds is that attacks that do less than the threshold do nothing.

This is not the same as, say, temporary hit points, which use a different wording (PHB, pg. 198):

When you have temporary hit points and take damage, the temporary hit points are lost first, and any leftover damage carries over to your normal hit points. For example, if you have 5 temporary hit points and take 7 damage, you lose the temporary hit points and then take 2 damage.

This talks about "carrying over" hit points. If that is how damage thresholds were supposed to work, similar wording would likely have been used. As it stands, it simply says that it "takes damage as normal", which means, from your example, the warship takes the full 50 points of damage.

From your own quote:

...in which case it takes damage as normal.

Damage of 29 vs object with threshold of 30? No damage.

Damage of 30 vs object with threshold of 30? 30 damage.

Damage of 50 vs object with threshold of 30? 50 damage.

This is different from something like the Feat Heavy Armor Master, where:

While you are wearing heavy armor, bludgeoning, piercing, and slashing damage that you take from non-magical weapons is reduced by 3.

The answer is in the rule that you quoted, but I'll reword it for you in case that helps.

It takes no damage if the total is below the threshold and it takes full damage if the total is greater than or equal to the threshold. So in your example it takes the full 50.