But where do you draw the line? How different can one person's perceptions be from another's before the god himself becomes a "different god"? I do understand what you're saying, and to some extent, I agree. But there is one god referred to as the "Abrahamic God." Most Christians see Him as a Trinity. The LDS concept of God differs somewhat from the mainstream Christian concept, but we still believe in the Father, Son, and Holy Ghost. We even believe they are "one God" -- just "one" in a different way than as defined by the Nicene Creed. Jews and Muslims believe in the Abrahamic God, too, but to them the idea that He has a Son is preposterous. I still think Christians, Jews and Muslims all believe in the same God. Once you get into Eastern religions, it gets a bit more fuzzy.
The line is debatable, but some things I think are pretty straightforward. I don't think they believe in the same god at all. It's not drawn right down the line between those three religions, though. For instance, some Christians and Muslims may have a more similar god concept than two drastically different Christians.
Perhaps an easy way to divide it is to think of it in turns of attributes and personality.
When it comes to attributes, the gods differ. Mormons, and correct me if I'm wrong, believe God the Father has a physical body, that he did not create everything in existence, and that he was once like man. This differs rather drastically from the concept of god being an eternal and unimaginable force that exists outside of spacetime that created all of existence, and for which nothing separate independently exists. This is an extremely large discrepancy in describing different deities. And, when most variants of Christianity are compared to Judaism and Islam, they believe God has a son, who is also part of the godhead, and exists as part of an eternal trinity. Muslims and Jews find this to be incorrect (and for some, blasphemous), and say that god is purely one thing with no separate persons at all. Again, this is a rather large discrepancy.
When it comes to personality, this is arguably the more practical difference. Some people believe god is the most loving thing in existence, full of patience and kindness. Some believe everyone goes to heaven, and/or that god really loves each and every human being. Others view god as a wrathful tyrant, who is willing to torture people infinitely, or infinitely separate beings from him and not allowing them to die. Other views that god along with this is that God is jealous, has committed or commanded genocide, or wants Sharia law, etc. And like I said, it's not drawn between Judaism, Christianity, and Islam, although that's there some of the lines are. Gods can differ within religions almost as drastically as they can differ between religions.
Basically, some people would read stuff like this from the Qur'an and say that's their god, while others would say it's the exact opposite:
004.056 Those who reject our Signs, We shall soon cast into the Fire: as often as their skins are roasted through, We shall change them for fresh skins, that they may taste the penalty: for God is Exalted in Power, Wise.
008.054 (Deeds) after the manner of the people of Pharaoh and those before them": They treated as false the Signs of their Lord: so We destroyed them for their crimes, and We drowned the people of Pharaoh: for they were all oppressors and wrong-doers.
008.055 For the worst of beasts in the sight of Allah are those who reject Him: They will not believe.
022.018 Seest thou not that to God bow down in worship all things that are in the heavens and on earth,- the sun, the moon, the stars; the hills, the trees, the animals; and a great number among mankind? But a great number are (also) such as are fit for Punishment: and such as God shall disgrace,- None can raise to honour: for God carries out all that He wills.
022.019 These two antagonists dispute with each other about their Lord: But those who deny (their Lord),- for them will be cut out a garment of Fire: over their heads will be poured out boiling water.
022.020 With it will be scalded what is within their bodies, as well as (their) skins.
022.021 In addition there will be maces of iron (to punish) them.
022.022 Every time they wish to get away therefrom, from anguish, they will be forced back therein, and (it will be said), "Taste ye the Penalty of Burning!"
040.070 Those who reject the Book and the (revelations) with which We sent our apostles: but soon shall they know,-
040.071 When the yokes (shall be) round their necks, and the chains; they shall be dragged along-
040.072 In the boiling fetid fluid: then in the Fire shall they be burned;
040.073 Then shall it be said to them: "Where are the (deities) to which ye gave part- worship-
040.074 "In derogation of God?" They will reply: "They have left us in the lurch: Nay, we invoked not, of old, anything (that had real existence)." Thus does God leave the Unbelievers to stray.
040.075 "That was because ye were wont to rejoice on the earth in things other than the Truth, and that ye were wont to be insolent.
040.076 "Enter ye the gates of Hell, to dwell therein: and evil is (this) abode of the arrogant!"
Gods that people believe in, even in a given religion or collections of related religions, can be more different than Gandhi and Stalin.