You can apply class abilities to spell casting from magical items
While casting from an item is not the same as a Casting a Spell action, it is still casting a spell. Unless a term is given a special definition in 5e game terms, you use the common definition for that term. You are casting the spell, the ability to do so comes from the item. For instance, the Staff of Fireballs:
.. you can use an action to ... cast one of the following Spells ...
Is no different, in this regard, to Wand of Fireballs from:
you can use an action to ... cast the Fireball spell ...
Jeremy Crawford has posted a number of answers we can point to in the time since this question appeared that verifies this really is the case. Each time he's been asked about a specific class ability the answer has been an affirmative. These were made when his tweets were official rulings, but since have been "downgraded" by the Sage Advice Compendium to his personal advice and perspective. While this is true, it still is clarification of the intent of the designers.
Metamagic
Can Metamagic be used on magic item casting a spell? Yes.
If a magic item's description says you cast a spell from it, you can use Metamagic on the spell. #DnD
and
Metamagic works w/ any spells that sorcerers cast. Wild Magic Surge can work w/ any sorcerer spell they cast. #DnD
Wild Magic
Can using a magic item cause a Wild Magic Surge? Yes
Yes.
Arcane Ward
Can casting from a wand/staff recharge Arcane Ward? Yes
Arcane Ward/Twinned Spell works when you cast a qualifying spell. It even works when an item says you cast one. #DnD
The Hexblade patron adds shield to the class list. This is a warlock patron offered in an Unearned Arcana playtest document, so your DM would have to allow playtest material.
If your warlock can somehow change patrons to Hexblade (not normally possible), then shield would be on your warlock's spell list.
Otherwise, if multiclassing is allowed in your campaign, one of the other PCs could take one level of warlock and choose the Hexblade patron to get shield.
Best Answer
This is a specific case that overrides the general one
Firstly, on PHB, p. 7:
Generally speaking, magic staves (and other magic items) that allow you to cast spells, usually "once per long rest" or "once per dawn", or through the use of expending charges, do not require that you have those spells in your spell list.
The Staff of Defense and Spider Staff, however, both override that by stating "if the spell is on your class's spell list" (LMoP, p. 53), as you note. Therefore, these staves do require that you have access to these spells already, which is different to other magic items that allow the casting of spells.
As @V2Blast points out, the Starter Set came out before the PHB or DMG were released, so such design conventions had likely not been finalized/refined at the time of its release. It may be that these are different from the items in the DMG because they decided to change how such restrictions were implemented, using attunement requirements rather than requiring you to already have access to the spells that a magic item provides.