The prefix "a" merely means "without" or "not". Thus, "atheism" signifies being without "theism". Then we have to define "theism". Even with the definition you provided, all one needs to be considered a "theist" is a belief in the existence of God or gods. Polytheists are theists, monotheists are theists, etc. (this I have to assume you agree with, for if this wasn't true, it would create insurmountable confusion in linguistics). So, "atheism" is being "without" theism or "without a belief in the existence of God." Thus, anyone who is "without a belief in the existence of God" can accurately be described as an "atheist".
Even if we were to use the entirety of your definition for theism including a personal God who interferes with mankind, it would act to include more people under the classification "atheist". This is easy to see because under your definition, since the prefix "a" means without (think "amoral" or "without morals"), "atheism" would include anyone without a belief in a personal God. Thus, deists, under your definition, would then be included under atheism. This, however, is incorrect because of the use of the word "especially" in your definition, signifying that the 2nd clause is unnecessary, but often used.
See how that would cause a huge problem?