What place do ontological arguments for God have in Christianity? As I understand it, the ontological argument that is sometimes applied to the existence of God goes something like this:


  God by definition is perfect. Non-existence would be an imperfection, therefore God exists. 


However it seems to me there is something a bit off with the argument. For example, one could say,


  I define a monster to be the scariest possible being. A monster would be scarier if it exists and is about to attack me. Therefore, I am about to be attacked by a terrifying monster.


Yet this version seems clearly false.


First of all, are these arguments actually equivalent or is there something fundamentally different between them that I missed?
Secondly, is this the same argument used by Christians or is there another variant? If it is/was used, who was it used by and how widely accepted is it? Do any major theological traditions raise any significant objections to this argument?
Regarding your first question, there is a key difference between your two examples. They would be more parallel if you framed the first one like this:


  God is the most perfect possible being that can be thought of. Non-existence or a lack of regular competition in barbecuing championships would be an imperfection. Therefore, God exists and regularly competes in barbecuing championships.


Neither competing in barbecuing championships nor attacks on you are essential to being a perfect being or the most scary being, respectively. However, the most scary being could be argued to be one that exists, rather than one that doesn't, so:


  A monster is the most scary possible being. A being that exists is scarier than one that doesn't exist. Therefore monsters exist. 


If we hold existing as absolutely essential to monsterhood over other possibly scary attributes, then this holds true. But, we may be able to think of things that seem scarier that don't exist (say a plague of Tribbles on Kronos) than those that do exist, so one might argue that existence isn't essential to monsterhood. 

I think Anselm makes a stronger argument towards God, however. In his rejection of Gaunilo's critique, he notes that talking about God is different than talking about the most perfect island. At the very least, we know that the definition of perfection is tied intricately to God in the Christian understanding of God, so whatever God is must be more perfect than what God isn't. 

As @justbelieve notes the real rub with the argument is that it cannot prove a particular deity, even if the argument works. We might find that the most perfect possible being -- with existence being the key attribute to that being's essence -- might be very much unlike what we want to label God in a hypothetical reality where God didn't exist. 

Speaking of hypothetical realities, Alvin Plantinga has a revised Ontological Argument utilizing modal logic that deals with the idea of multiple realities. Perhaps someone who is a braver soul than I would like to explain it. 

Regarding your second question, the Ontological Argument's major proponents have included St. Anselm, René Descartes, Karl Barth and Alvin Platinga, just to name a few. Contra the argument within Christian tradition would be a great many others, including St. Thomas Aquinas (cf. Summa Theological 1.2, especially questions 1 for the rebuttal and 3 for his alternative).