Is my player doing AC right? (2014 5e rules) Help please.

I want to lead with this is a mechanics question, this is a great player in roleplay and person, I'm just not sure on these mechanics.

I have a multi-classed player
1 Level Barbarian
2 Levels Paladin
5 Levels Wizard

They are using the barbarian AC calculation of unarmored defense

• 10 + Dexterity modifier + Constitution modifier

They also subclassed into Bladesinging which has Bladesong grant the following buff

• You gain a bonus to your AC equal to your Intelligence modifier (minimum of +1)

So frequently their AC is

• 10 + Dexterity modifier + Constitution modifier + Intelligence modifier

Now I just found that Armor Class in the player's handbook says
"Your base AC calculation is 10 plus your Dexterity modifier. If a rule gives you another base AC calculation, you choose which calculation to use; you can’t use more than one."
https://www.dndbeyond.com/sources/dnd/br-2024/rules-glossary#ArmorClass

I have one person telling me that the player is crossing AC bonuses from multiple classes and this rule says you can't do that, I have another person saying the Bladesong isn't base AC and that the player is playing perfectly.

Thoughts?