The mentions of Wizard spells on page 97 are EXCLUSIVELY using the phrase "Wizard Spell List".
This is not true.
The 10th and 11th line on page 97:
"When you reach fighter level 10, you learn an-other Wizard cantrip of your choice"ypi
IMO the term "Wizard cantrip" here on page 97 means the same thing when that term is again used on page 98 under War Magic:
"....you can replace one of your Wizard cantrips ...."
IMO this means the same thing in both places, as you noted it does not "change meaning"
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".
Yes and when it talks about learning a "Wizard cantrip" it is refering specifically to the spells being learned from the Eldritch Knight subclass. If your EK has multiclass levels in Wizard and Warlock and Sorcer you do not get to chose another "Wizard cantrip" for those classes when you gain 10th level in Eldritch Knight. You only choose it for Eldritch Knight. It only refers to cantrips for Eldritch KNight.
The fact that they used this term "Wizard cantrip" to me means this is what they are talking about every time they use the term - the spells you get from the Eldritch Knight feature
So it cannot possibly mean that, and it must mean "a cantrip from the wizard spell list"
So if I have a Warlock 1, Wizard 1, Sorcer 1, Eldritch Knight 9 and I gain 10th level in Fighter I can get a cantrip in all 4 of those classes? Because that is what you are saying if it means the same thing in both places. Either the term "Wizard cantrip" is meant to mean the Cantrips from the EK feature or it is meant to mean all your cantrips from all classes that are on the Wizard list.
Of course this is not what it means! In my opinion it is clear it is referring to the same thing both times the phrase is used and in my opinion it means the same thing in both places..
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".
It is not in my opinion. It is referring to a spell you get through the EK feature specifically (which also happen to be on the Wizard list), but I don't believe it refers to all spells that are on the Wizard list for all classes you have.
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.
That is not true, it includes the new spell you get when you pick up that feature. It does not include spells from other multiclasses and you can't get new spells from the "Wizard list" on those multiclasses when you gain a level in Eldritch Knight.
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.
As I stated earlier RAW is unclear. In my opinion RAI is for them ONLY to work with spells gained through the Eldritch Knight subclass feature. In context I think that is the clear intent. You apparently disagree.
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.
You said it not me: It is written in a way as to assume you have no spells from another class. That it means it was not written with the INTENT to apply to other classes. Other classes were not considered, it was written with the INTENT to apply to a single class Eldritch Knight and I am talking about rules as intended here.
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"
Maybe eventually we will see something in Sage advice, but my opinion on what the designers intended is well thought out and it is my opinion.