Does multiclassing suddenly allow you to cast spells from Magic Initiate (or other similar features/feats) using more spell slots?

The following question already exists and explains that you can cast spells gained from the Magic Initiate feat if and only if the class you choose is the same as one of your own:

  • Can you cast a spell learned from the Magic Initiate feat using spell slots?

So let’s assume a Wizard take the Magic Initiate feat and learns the healing word spell with the Bard option of the feat. It is established already that they cannot cast this spell using their own spell slots, for some more supporting evidence classes received errata and the Wizard class now states the following under “Spellcasting” (the bolded word “Wizard” was added):

[…] The Wizard table shows how many spell slots you have to cast your wizard spells of 1st level and higher. […]

With that established, I’m a bit confused what happens when this Wizard decides to multiclass into Bard; the multiclassing spellcasting rules state:

[…] 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. […]

There is not longer a way to distinguish between Wizard spell slots and Bard spell slots. To me this implies the following:

Before multiclassing the Wizard could not use any of their spell slots to cast healing word, but by gaining one level of Bard they can now use all of them. Is this truly the case or have I missed a rule somewhere?