The first way is correct, as further evidenced by the introduction of the Artificer in Eberron: Rising from the Last War.
The optional artificer multiclassing rule for spell slots says:
Add half your levels (rounded up) in the artificer class to the appropriate levels from other classes to determine your available spell slots.
First, this confirms that at the very least, when one of your half-caster classes is Artificer, you divide your levels in Artificer, then round them up, before adding them to anything else. If you are a multiclass Artificer/Paladin, the second way listed in the OP (add first, then divide and round) is simply incompatible with the rules for artificer. The rules listed here for artificer demonstrate that the intent is that the dividing and rounding occurs before adding the results together.
To be clear, this is my argument: the Artificer's multiclassing rules are improving the Artificer's implementation of an existing mechanic not introducing a new mechanic entirely. If the correct ruling was "add first, then divide and round", then the Artificer's rules represent a completely new method that we have to add on top of the existing one. If the correct ruling is "divide and round first, then add", the Artificer's rules just changes the direction of the rounding in the already established mechanics for mutliclassing half casters.
I argue that change to an existing mechanic is more likely to be the correct ruling than a new mechanic that must be considered in addition to the original. And you may disagree. But I am adding to the body of evidence for a particular ruling that in this answer was only given JC tweet evidence.