Some competing definitions.
If one is to define 'religion', one should look at the characteristics of every religion, even those that have no god or gods (such as the Church of No God, Buddhism, and to a lesser degree, Shintoism).
One characteristic is that all religions, regardless of that they are, begin with a circular argument (otherwise known as an argument of faith).
It is more than this, however. A religion begins with this, but then extends that argument with other arguments.
For example, in Christianity, the initial circular argument (and the argument of faith) is that Christ exists, and that He is who He says He is, namely the Son of God. ALL other arguments in Christianity stem from this initial argument, including the belief of resurrection.
In Buddhism, the initial circular argument is that Buddha was the wisest of men. ALL other arguments (including the belief in reincarnation) stem from that initial argument.
Islam believes in God, but not Jesus Christ. It is the same as the Jews. The difference with the Jews is that they view Jesus Christ merely as a great teacher, but is not the promised Messiah. Most Jews today are stuck in the Old Testament, but give it just lip service. Islam is also stuck in the Old Testament, concentrating on the wrath of God, and justifying their harsh judgement on anything not Islam.
Shinto's initial circular argument is that objects (particularly unusual objects) contain a spirit, or demon...some friendly, some evil, some simply mischievous. This is also the religion of many American indians. The construction of little temples, or shrines, is commonplace through Japan. You'll see them tucked away on a street corner even in the cities.
The initial circular argument of the Church of No God is that there is no god or gods or spirits at all. ALL other arguments stem from this initial circular argument.
Now let's look at some of your quotes:
Paul Tillich has it basically right. Religion describes how the world works for the believers. It gives meaning to ones life.
Max Stackhouse is also basically right. He is describing the circular argument, or the argument of faith here, and how it's a basis for all religions.
Emile Durkheim is only partially right. Religion is a unified set of beliefs (all stemming from that initial circular argument) and it defines what is 'sacred'. Emile is wrong in that religion does not have to be organized.
Sigmund Freud is completely wrong. As is typical for him, he describes it as a 'childhood neurosis'. You might say Freud invented psychoquackery.
Karl Marx believed in the Church of No God. He describes any nonbeliever in that religion to be 'on opium'.
This brings up the concept of fundamentalism, which can exist in any religion.
Since the circular argument in and of itself is not a fallacy, and in and of itself is not a proof, any such argument can only be taken at face value. It may be True, it may be False. It is not possible to prove a circular argument either way. It is not possible to prove whether any god, gods, spirits, demons, etc. exist or not.
If one attempts to prove a circular argument, they commit the circular argument fallacy. This is what a fundamentalist does. Most attempt to try to prove their religion True, even though it's not possible.
Like any fundamentalist does, Karl Marx is simply trying to condemn anyone that is not of his religion and to prove his religion is True.
Richard Dawkins also is a believer in the Church of No God. He is quite fundamentalist. It is not possible to prove that no god or gods exist. Like Freud before him, he condemns non-believers with essentially name-calling.
Frederich Nietzsche is just trying to make a semantics fallacy, pretty common today as well, especially among Democrats.
Vladimir Ulyanov, who later changed his name to Lenin, belonged to the Church of No God. To him, the State should be God. He describes religion as a 'yoke', and then implements a heavy yoke upon the people of Russia.
This is the same belief of many Democrats, including Joe Biden.
