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Considering that Saber Lily is simply Saber with a different clothing that leans on the more "girlish" side, she has a tragic background (being betrayed) that made her regret her decision that the reason why she become a Heroic Spirit is to change her past.
If you were to see that, it feels that you want to protect her and support her emotionally; sure she is capable on annihilating an enemy such as other Servants but what she needs is emotional protection and support.
The thing about Lizlet is that she is already capable of protecting herself (the manga and anime shows that she has great physical strength despite that she appears to be the opposite) and her only flaw is that she gets intimidated fast (she is the comic-relief in the main cast of Omamori Himari) and unless you attack her true body which is a century-old teacup, she is pretty much invulnerable... well her human manifestation.
Saber Lily on the other hand, like the other servants, can be harmed and even killed without attacking a specific object and I do not like the idea of seeing a 2D girl being hurt and even swimming in their own blood; there are pictures of original Saber being in this condition but seeing Saber Lily in that condition is more heart-breaking.
Though I also did have some sort of attachment towards another girl which is Nodoka Haramura, it was Saber Lily that ultimately got my affection.