Do Invocations Count As Warlock Spells

Apologies for the newbie question, but Pact of the Tome specifically states that its spells count as warlock spells, but the other invocations that grant spells don't specify that.  For example, Silent Image is not normally on the warlock spell list, but it can be obtained via warlock invocation.  Does that make it a "warlock spell" for other purposes?

A strict reading of the rules would say no, since there's nothing there saying it makes it a Warlock spell. There's a rule saying that if a feature grants a spell as "always prepared" then it counts as a Warlock spell, but the invocations (kind of pointedly) do not use the term "always prepared".

The problem with this is that technically if they're not Warlock spells, there's no rule telling you which spellcasting ability to use for them, or enabling you to use a spellcasting focus with the ones that have material components, since both of those parts of the Pact Magic feature explicitly only apply to Warlock spells. Thus it seems extremely unlikely that that's the intended interpretation.

In my experience, regardless of what the rules technically say, pretty much everyone just treats them as though they count as Warlock spells and calls it a day.