Chaosmancer
Legend
To me it is pretty clear what the intent is and the intent is for it to apply to the spells you received from the subclass. When they use those phrases on page 97 and at the top of page 98 it is clear they are talking only and specifically about the spells gained through EK. Those areas (under cantrips and Changing Tour prepared spells respectively) CAN'T refer to any other spells and absolutely do not refer to spells received from another class. If those phrases in these two sections do not "change meaning" then they would still refer to the spells received from the subclass later under War Magic and Improved War Magic on page 98.
The mentions of Wizard spells on page 97 are EXCLUSIVELY using the phrase "Wizard Spell List". There is not a single part of that text that uses the phrase "wizard spell". The only mention of a "wizard cantrip" is following the text on learning cantrips from the Wizard Spell List, and is for the level 10 spot where you learn a new "wizard cantrip". To have that mean "Fighter Cantrip" instead, would mean that the level 10 cantrip ability is allowing you to learn a cantrip you received from the class... which you would already know. So it cannot possibly mean that, and it must mean "a cantrip from the wizard spell list"
And the same logic must apply here to the top of 98 where you change your prepared spells. The phrase "with another Wizard spell for which you have spell slots" must absolutely be referring to "a spell on the wizard spell list". If it meant "a wizard spell you gained via this subclass" like you keep saying then you could only swap spells to... spells you already know... which does nothing.
So, the only argument you could possibly be making here is that War Magic and Improved War Magic don't work with cleric/druid/warlock/bard spells/cantrips. Which... fine, but then we run into the Innate sorcerery problem.
What you wrote that I bolded in red above would also support this interpretation. If it is written to assume that you do not have any other spells except subclass spells then that also means by definition they did not "intend" it to apply to spells from other classes when they wrote it. They "intended" it to apply to the spells you received from EK.
No, that is not a good assumption. Because the phrasing used on learning spells is generally identical across all classes. That would mean that they never intended something like Wall of Force to be used with Quicken Spell... but obviously that is intended because of how the sorcerer class functions.
RAW Initiate Sorcery clearly only works on Sorcerer spells and the multiclass Sorcerers I have seen played in 2024 do not get to use that feature on species spells, feat spells or spells from other classes. If you let it work on all spells (or even on all spells on the Sorc list) then the optimal caster would all have a level of Sorcerer.
Eldritch Knight wording is a bit more ambiguous RAW, but I don't think it is ambiguous RAI. If the "intent" is to allow spells and cantrips from other classes then you are saying that it is intended that a multiclass 17th level Warlock-Eldrtich Knight should be able to let loose with 4 Eldritch Blasts followed by up to 4 more attacks at high level, without action surging and adding 8x Hex damage and 4xCharisma damage. I do not think that is "intended".
I think you are taking a far too limited read on this, and trying to put forth a very narrow potential based solely on them clarifying that the Fighter is using Wizard spells and not Fighter spells (which don't exist) and that they did not write "a spell from the wizard spell list" when talking about changing your prepared spells. Which would have been needlessly wordy. And also still does nothing to prevent wizard spells gained from feats to function normally, because they are still wizard spells and that must mean "spells from the wizard class list"