I know the errata has cleared this up but I never read the multiclassing rules that way in the first place.
In the Wizard section, it says, "Your spellbook is the repository of the wizard spells you know, except your cantrips, which are fixed in your mind."
In the multiclassing section, it says, "You determine what spells you know and can prepare for each class individually, as if you were a single-classed member of that class. "
If you are a Wizard 1 / Bard 5 then your spellbook, which contains all the spells you know, can only contain spells that a Wizard 1 knows. A Wizard 1 can't know any level 3 spells therefore they can't put any level 3 spells in their spellbook. At least, that's the way I always read it.