Extra Spell is specifically addressed in the FAQ (p. 40):

Can you take spells from spell lists other than your own with the Extra Spell feat (CAr 79)?

The Extra Spell feat allows you to choose a new spell, but it does not remove the restrictions of how you would normally pick your spells—so they must be picked from your own spell list.

This is the general rule -- you have a class ability that lets you cast spells from a particular class list only. For instance, the wizard has the following ability:

Spells: A wizard casts arcane spells which are drawn from the sorcerer/ wizard spell list.

For something to give you spells not on your class list, it has to explicitly call this out as an exception to the more general rule. The Master Specialist ability is worded almost identically to the wizard's regular spell book entry, so it clearly does not allow this:

When you reach 2nd level, you can add one spell of your chosen school to your spellbook. The spell can be of any level that you can cast, and it is in addition to the normal spells gained when increasing your level.

If it isn't on your class list, how would you even determine if it was a level that you can cast?

Extra Spell is specifically addressed in the FAQ (p. 40):

Can you take spells from spell lists other than your own with the Extra Spell feat (CAr 79)?

The Extra Spell feat allows you to choose a new spell, but it does not remove the restrictions of how you would normally pick your spells—so they must be picked from your own spell list.

This is the general rule -- you have a class ability that lets you cast spells from a particular class list only. For instance, the wizard has the following ability:

Spells: A wizard casts arcane spells which are drawn from the sorcerer/ wizard spell list.

For something to give you spells not on your class list, it has to explicitly call this out as an exception to the more general rule. The Master Specialist ability is worded almost identically to the wizard's regular spell book entry, so it clearly does not allow this:

When you reach 2nd level, you can add one spell of your chosen school to your spellbook. The spell can be of any level that you can cast, and it is in addition to the normal spells gained when increasing your level.

If it isn't on your class list, how would you even determine if it was a level that you can cast?

Extra Spell is specifically addressed in the FAQ (p. 40):

Can you take spells from spell lists other than your own with the Extra Spell feat (CAr 79)?

 

The Extra Spell feat allows you to choose a new spell, but it does not remove the restrictions of how you would normally pick your spells—so they must be picked from your own spell list.

This is the general rule -- you have a class ability that lets you cast spells from a particular class list only. For instance, the wizard has the following ability:

Spells: A wizard casts arcane spells which are drawn from the sorcerer/ wizard spell list.

For something to give you spells not on your class list, it has to explicitly call this out as an exception to the more general rule. The Master Specialist ability is worded almost identically to the wizard's regular spell book entry, so it clearly does not allow this:

When you reach 2nd level, you can add one spell of your chosen school to your spellbook. The spell can be of any level that you can cast, and it is in addition to the normal spells gained when increasing your level.

If it isn't on your class list, how would you even determine if it was a level that you can cast?

Extra Spell is specifically addressed in the FAQ (p. 40):

Can you take spells from spell lists other than your own with the Extra Spell feat (CAr 79)?

The Extra Spell feat allows you to choose a new spell, but it does not remove the restrictions of how you would normally pick your spells—so they must be picked from your own spell list.

This is the general rule -- you have a class ability that lets you cast spells from a particular class list only. For instance, the wizard has the following ability:

Spells: A wizard casts arcane spells which are drawn from the sorcerer/ wizard spell list.

For something to give you spells not on your class list, it has to explicitly call this out as an exception to the more general rule. The Master Specialist ability is worded almost identically to the wizard's regular spell book entry, so it clearly does not allow this:

When you reach 2nd level, you can add one spell of your chosen school to your spellbook. The spell can be of any level that you can cast, and it is in addition to the normal spells gained when increasing your level.

If it isn't on your class list, how would you even determine if it was a level that you can cast?