Because one believes in one god does not make that one god the same god. The word "god" is just that. In different languages there are different words for god - Allah is one of the many.
JHVH, the Hebrew description of The One True God (I AM THAT I AM), references the one God without a name. He just IS. In the flesh, JHVH is called Emmanuel and Jesus. After Jesus was spotted walking on water by the apostles he said:
Matthew 14:27 Young's Literal Translation (YLT) "... and immediately Jesus spake to them, saying, `Be of good courage, I am [he], be not afraid.'"Jesus self identifies as I AM as can be attested to in this literal translation of what Jesus said. Be clear - Christianity, although it refers to the Holy Trinity, is monotheistic! The Holy Trinity are the three aspects of JHVH, and all aspects are God. The word "three persons" is misapplied. God, except in his existence as Jesus Christ, is not a person - He is God!
In order to be a true Christian theist, one must worship JHVH as Holy Scripture presents Him. If a theist misrepresents the God of Holy Scripture, that person is not a Christian because God is more than a word - he is truth. God identifies himself quite well in scripture, and to misidentify Him creates another god more suitable to the liking of the theist.
What separates Christians from the rest of the pack is doctrine. Christians are theists who actually believe in and live by the teachings of Jesus Christ. Most people in the western world claim to be Christians. Claiming Jesus as savior is a mockery unless one actually does God's will because Jesus is God! What is His will? That his creation be like Him since we were made in His image. What does that mean? God is love; we too are to love, both God and our fellow man. Therefore, a theist is anyone who believes in the existence of God. Almost everybody is a theist - it's just a difference in the identity of God.
Muslims believe in Jesus as well. In the Q'uran, he is transliterated Isa. However, their use of the word "Jesus" is not based on Holy Scripture. Mohammad added to scripture with his own writings. His Jesus never died on the cross and was a mere prophet. Muslims are theists and use Judeo-Christian names, but their Allah and Isa are what biblical Christians would call "anti-christs". The character of their god and prophet is much different than what Christian theists believe. Those who make them the same, are Unitarian and Universalists in theology, and are thus anti-Christians.
Deism came from the Latin word deus which means "god". As one can see, both theism and deism are beliefs in god. Just as there are differences in theism, deism is much different than theism although they share the same thought. In fact, deism is theism because they too believe in god. Unlike theism though, which allows for pantheistic beliefs, deism is monotheistic - there was one creator god, and then he removed himself from nature. Hence, in deism, there is no personal God which deals with individuals, leaving mankind to fend for themselves.
The appearance is that deists who live their daily lives without God are atheists because their dependence is on mankind themselves. Likely, a better word for deists is "selfists" because those are people who belong to a group which effectually self-worships. Their theism is that their own morality is the salvation of mankind. The difference is that deists acknowledge a Power which they often refer to as Providence, but is not the JHVH of Holy Scripture. Many of our founding fathers in the United States were deists, and believed in Providence, but not JHVH. Since their manner was reason, their dependence was on their own self. Although they acknowledged a higher power, they were not truly theists because in daily living and practice there is no God.
Deism is the unknown god that Paul talks about in scripture. That religion is philosophy - the love of wisdom. Whose wisdom? Mankind's! Philosophers are effectually selfists. Modern day psychology is the application of philosophy to improve the mental state of men. It's origin comes from philosophy and mostly eastern paganism. The focus is on self-improvement - balancing out characteristics within which is the yin and yang of balancing forces. Many Christians buy into the philosophy and depend on psychology ( the study of the soul) rather than God's healing power. In essence, they too are selfists.
Atheism is without belief in god(s) of any type. They are the more honest of those outside Judeo-Christianity. They admit their unbelief. Many agnostics are atheists because they have yet to be convinced that there is a god or gods. They stand ready to be theists, but it is an impossibility that they can ever be without a change in belief. Why? Because theism is not scientific; it is by faith. In fact, science too requires even more faith because of the lack of the ability to scientifically prove or disprove the existence of god(s). Agnosticism means skepticism, but they are biased toward unbelief. Therefore, since they depend on reason, they are more atheists than deists because in their viewpoint, there was not a creator.
In boils down that all isms are selfists because they all depend on their own wisdom. Ever since Adam used wrong judgment, the other gods (theists) are always themselves. Even when others are Buddhists, Muslims or any other religion, they are selfists because they are believing what is right in their own eyes, contrary to the truth associated with JHVH.