Multiclass Paladin + Primary Spellcaster = What paladin spells can I prepare and cast?

I’m sure this has been asked somewhere already, but I couldn’t find it.

Let’s say I’m Paladin 5 / Warlock 7. https://www.dndbeyond.com/sources/basic-rules/customization-options#Spellcasting

When I choose what Paladin spells I know, I consider myself as a single-classed Paladin 5, which would have 2nd level spell slots, so I can know 2nd level Paladin spells.

Spells Known and Prepared. You determine what spells you know and can prepare for each class individually, as if you were a single-classed member of that class.

When I consider what spell slots I actually have, I look on the multiclass spellcaster table, and see that I only have 1st level Paladin spell slots.

Spell Slots. You determine your available spell slots by adding together all your levels in the bard, cleric, druid, sorcerer, and wizard classes, and half your levels (rounded down) in the paladin and ranger classes. Use this total to determine your spell slots by consulting the Multiclass Spellcaster table.

So, if I want to cast one of those 2nd level Paladin spells, I’ll need to use one of my 4th level spell slots from Warlock.

Is all of that correct?