There is no rule for oversized weapons.
The section that you quote is not rules text. Rather, this section that refers to oversized weapons is part of the instructions for creating a custom monster:
Creating a Monster
The Monster Manual contains hundreds of ready-to-play monsters, but it doesn’t include every monster that you can imagine. Part of the D&D experience is the simple joy of creating new monsters and customizing existing ones, if for no other reason than to surprise and delight your players with something they’ve never faced before.
Creating a Monster Stat Block (Steps 11-15)
Step 11. Damage
A monster’s damage output—the amount of damage it deals every round—has a direct bearing on its challenge rating, and vice versa. You can determine a monster’s damage output in one of two ways.
Base the Damage on the Weapon. Alternatively, you can use a die expression to represent the damage that a monster deals with each of its attacks based on whatever weapon it is using.
Big monsters typically wield oversized weapons that deal extra dice of damage on a hit. Double the weapon dice if the creature is Large, triple the weapon dice if it’s Huge, and quadruple the weapon dice if it’s Gargantuan. For example, a Huge giant wielding an appropriately sized greataxe deals 3d12 slashing damage (plus its Strength bonus), instead of the normal 1d12.
Again, this is not rules text. These are instructions for creating your own stat blocks, and gives some insight into how the authors typically selected damage dice when designing monsters. But this is not a rule to be invoked at the table, and the authors didn't even follow it all of the time when designing monster stat blocks. For example, the fiend Baphomet, size Huge, carries a glaive named Heartcleaver:
Baphomet wields a great glaive called Heartcleaver.
Heartcleaver's attack is described:
Heartcleaver. Melee Weapon Attack: +17 to hit, reach 15 ft., one target. Hit: 21 (2d10 + 10) force damage.
Since Baphomet is a Huge monster, we would typically expect the weapon attacks to deal 3dX, but Baphomet's do not. So this bit about oversized weapons is not a rule. It's a design principle that was usually followed, but not always.
So the answer to your question is "do what enlarge says to do", and there are no other rules to consider. So you get an extra 1d4 damage.
We can further motivate this ruling by observing that I as a player should be able to figure out what the spell does without having to read a section about creating custom monsters from the Dungeon Master's Guide that was written for the DM’s eyes only. The spell enlarge/reduce is printed in the Basic Rules and System Reference Document, documents which do not contain the material about homebrewing monsters from the DMG. We should be able to figure out what the spell does even if all we have is the Basic Rules or SRD, by reading the spell description and the rules for spells.