Some have argued that rather than define atheism as the absence of belief in gods, it makes more sense to define atheism as not worshipping any gods. There are advantages and disadvantages to this idea. A principle advantage is the fact that there is no need to go through contortions when it comes to apparently "real" gods, like a tree or Caesar. A principle disadvantage is the fact that it allows for a person to be an atheist but to believe that some god exists — an apparent contradiction.
Why is there an advantage in focusing on what atheists do or do not worship rather than what they do or do not believe? Most alleged gods are easy both to not believe in and to not worship — the existence of gods in religions like Christianity, for example, aren't so obvious that it's impossible to not believe that they exist. If you don't believe they exist, however, then not worshipping them is easy. One is follows automatically from the other.
With some alleged gods, however, their existence is a bit harder to completely ignore even if not worshipping them is normal. People have believed that gods have inhabited inanimate objects like trees or rocks. People have also believed that other living people have been gods, for example the Caesars in ancient Rome. Pantheists believe that the universe itself is God. Trees and rocks exist. Caesar existed. The universe exists. How can atheists disbelieve in them? We can't, but we're obviously still atheists — and of course other theists don't believe in these "gods," either.
They key is to argue that while trees, rocks, and the universe exist, they have not possessed qualities which justify calling them "gods." This requires having some set of characteristics which a god must have, itself a potential problem which can lead to complications down the road. All of this is alleviated if we simply define theism as worshipping at least one god and atheism as not worshipping any gods. It also eliminates problems with people claiming that atheists worship other things, like themselves or money.
How people behave is arguably more indicative of what they really feel inside than what they say, so perhaps focusing on behavior like worship would allow for a better, cleaner, and more accurate separation between atheists and theists. Even if "worshipping gods" is not the behavior we should be looking most closely at, perhaps we should be looking more closely at some behaviors than at people's beliefs.
This redefinition of atheism as not worshipping any gods doesn't come without costs, however, and the most significant is the fact that it will lead to counter-intuitive results in some cases: people we'd normally classify as atheists but who accept the existence of some gods and people we'd normally classify as theists but who are now labeled atheists because they don't worship any gods. The more such examples there are, the less useful this distinction may really be for describing reality.
As to the first case, it's a direct consequence of the aforementioned advantage: since defining atheists doesn’t focus on belief anymore, it allows for saying that someone is an "atheist" because they don't worship the universe or Caesar, but they are a "theist" who regard the universe or Caesar as being a god on some level. An atheist who accepts the existence of a god doesn't seem to make much sense.
The second case can occur because theism is normally defined as simply believing in the existence of at least one sort of god, but belief does not necessarily lead to worship. It's certainly what happens in most cases, but one does not necessarily or automatically follow from the other. If we define theism according to worship, then, someone who was at one time considered a theist would not be considered an atheist despite believing in one or perhaps even many gods which they simply don't bother to worship. That doesn't seem to make much sense, either.
Because of these contradictions, I don't think that it would be helpful to pursue trying to redefine atheism according to worship instead of belief. However, because of the advantages, I think it's important to acknowledge that this suggested redefinition at the very least offers some important insights into not only the nature of atheism and theism, but also the difference between the two. The lack of worship may not be the key factor that separates atheists from theists, but it is one which often separates them and which we should perhaps pay a bit more attention to.
