Eldritch Knights can only cast Abjuration and Evocation spells. But the list of spells available to choose from includes spells from all schools of magic.
The Eldritch Knight can choose from any Wizard school for the spells selected at 3rd, 8th, 14th and 20th level. So only those 1-4 spells can come from any Wizard school. All the other spells they choose have to come from Abjuration or Evocation. This is a bug in the system.
In fact, in the Errata: PLayer's Handbook v2.0 (2018) is specifically states :
Spells Known of 1st-Level and Higher (p. 75). The final sentence now ends as follows: “… unless you’re replacing the spell you gained at 3rd, 8th, 14th, or 20th level from any school of magic.”
So it's calling out that only the single spell can be replaced for a non-Abj/Evo spell, not your entire spell book.
Eldritch Knights can only cast Abjuration and Evocation spells. But the list of spells available to choose from includes spells from all schools of magic.
Eldritch Knight can learn spells from any school of magic at 8th, 14th, and 20th level.
And when they pick the archetype (level 3)
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That is partially correct.
The Eldritch Knight can choose from any Wizard school for the spells selected at 3rd, 8th, 14th and 20th level. So only those 1-4 spells can come from any Wizard school. All the other spells they choose have to come from Abjuration or Evocation. This is a bug in the system.
In fact, in the Errata: PLayer's Handbook v2.0 (2018) is specifically states :
Spells Known of 1st-Level and Higher (p. 75). The final sentence now ends as follows: “… unless you’re replacing the spell you gained at 3rd, 8th, 14th, or 20th level from any school of magic.”
So it's calling out that only the single spell can be replaced for a non-Abj/Evo spell, not your entire spell book.