Yes, they are both atheists, but they are still different positions. There is a difference between saying "I don't believe there is a God" and "I believe there is no God" - it is the same as saying "I don't believe the number of grains of salt in the ocean in even" and saying "the number of grains of salt in the ocean is odd"; or the difference between finding a defendant in the murder case to be "not guilty" rather than finding them "innocent". The definition of atheism encompasses both positions in the same way that the definition of theism encompassed both monotheism and polytheism, but that doesn't mean they are the same position.
The position "I don't believe in a God" is a response to the question "Do you believe there IS a God", whereas the position "I believe there is no God" is a response to the question "Do you believe there IS NOT a God". The positions are, themselves, subdivisions of the whole under the heading of disbelief in God (as both positions would give the same answer to the first question), but answering question one as quoted does not require or preclude the quoted answer to question two. You may not believe the number of grains of sand in the ocean is even, but that doesn't require you to believe the number of grains of sand in the ocean is odd.