Is Mindfulness the same as having an Observer?
The Observer is the experience of being aware of myself. It feels like I am witnessing my own thoughts and actions.


In Buddhism, "myself" as an entity would be a mind-object: a concept, something one conceived / imagined, which receives attention -- a 'self' is taught to not be in the range of actual experiences such as a touch, a smell or primitive mental phenomena (such as a mental image, or mental sound) that can be contacted.

In Buddhism, one is "aware" of the detailed micro experiences (that, among other things, give rise to a concept of self). In buddhism, this "detailed awareness", or "attention" to a specific "primitive experience" is called consciousness (vinanna), which is formulated in buddhism as six-fold: eye-consciousness, ear-consciousness, nose-consciousness, tongue-consciousness, body-consciousness, mind-consciousness1.

Then, it's said that if there's  eye organ and visual forms,  eye-consciousness arise. The coming of the three is contact. For example in my visual field now there are bed sheets, wall frames and a TV. But while I write this, I'm not conscious of any of these forms, only of this visual character I'm typing right now.

For example, I just realized there was a lamp in the ceiling with lights on in my visual field, even though it has been reachable all the time. Until that moment, there was no contact (my consciousness was not directed) to this particular visual object.

The same occurs with mind as a sense: there are mind objects (e.g. thoughts) a "mind organ", and a mind-consciousness arise; the coming of the three is contact (e.g. when one gives attention to a particular thought). With contact, pleasant, unpleasant (or neither one) occur -- and so on.


  Mindfulness is the process of being aware of what I am doing. (To me, this does not require an Observer, but that is what I am asking.)


I think this is close to what  mindfulness (sati) is in buddhism. It has different/related meanings depending on context but without going into details, I would summaryze as Thanissaro Bhikku put it: "What does it mean to be mindful of the breath? Something very simple: to keep the breath in mind."


  "Being aware of myself" means... Well, that I know what I am doing right now. Different from the other two.


I'm not sure how your "being aware of what I am doing" is different than your "know what I am doing".


  A Mind - is something that could do otherwise, it exercises choice on some level.


In buddhism, I think this could be mano, sankhara, or even possibly cetana.


  Awareness is the action of a sufficiently developed mind.


I don't quite understand what you mean on this one.


  Consciousness is awareness of being a self.


I would not say one is aware of a self (or 'myself', see first point) -- e.g. as if 'self' was an experienced object grasped by any of the 6 sense faculties. But one does elaborate a (by buddhism, incorrect) understanding of being or having 'a self'.

Now, to the question...


  Is Mindfulness the same as having an Observer?


In Buddhism (and if you understand mindfulness to be sati), no. 

For a detailed investigation on the meanings of mindfulness I suggest  Tse-fu Kuan's Mindfulness in Early Buddhism



1: The point, I think, is not to say that there is such a thing as consciousness and it can be perfectly divided in six part (an ontological statement). Rather, it is that understanding the mind in this way is useful / helps with training and understanding of dhamma, and consequently, with final liberation.