No, the spell is not a known spell on your list
The invocation merely grants you the ability to cast it at will - but it is not considered one of your 'spells known'. The language you quote simply states that you may cast it at will. But the qualifier you also quote remains:
cast mage armor on yourself
That is the only way you can cast it - but you can cast it at will. This is very helpful in cases of dispel magic or other means that may end the spell before you'd like it to end. It also doesn't use up a spell slot to maintain your mage armor, which is a huge benefit - especially with the limited slots a Warlock has.