No, the will to live does not constitute existentialism. It is part of it, but not the whole. There are various forms of existentialism, but one thing that unites them is that they are all responses to the loss of fundamental principles of life taken for granted in previous ages. Specifically, they are responses to the loss of principles concerning God and morals. Philosophy is not monolithic, of course, and so no single thesis was ever accepted by anyone. The mainstream of Western philosophy, however, had largely accepted that God existed, that God was understandable enough to be demonstrable and rationally believed in, that there were such things as moral facts, and that God had some role in justifying ethics.
Confidence in these views then started to break down due to advancements in both philosophy and science. People started to realize that the philosophical and scientific revolutions started by Galileo, Bacon, Descartes, and Newton were leading inexorably to the decline of traditional Western moral and religious beliefs. The ascendancy of Hume’s philosophy, which followed earlier views to their logical ends, was particularly influential on this score. Kant attempted to stop this, but his system turned out to be in no better position than those he sought to replace. His attempt to make God a transcendental condition of morality just made people doubt both God and morality even more.
These problems led to what might be referred to as the problem of the absurd: people are left attempting to live in ways that they have inherited from the past despite the fact that the belief systems justifying those ways of life have been discarded in favor of belief systems that entail the impossibility of meaningfully living in such ways. This in turn gave rise to the philosophical pessimism of Schopenhauer and others. One of their central theses was that life has no inherent meaning or value. Only God could instill life with such things, they argued, but they did not believe that God exists. Furthermore, they could see no other possible source of meaning or value. Thus they concluded that life had no meaning and an overall negative value. Life is not worth living.
Existentialists accept the problem of the absurd, but they reject the pessimistic conclusion that life is not worth living. Thus we can see how the will to live is an important characteristic of existentialism. It is not the whole of it because the will to live is shared by those who reject the problem of the absurd altogether. Such people typically do not focus on the issue, however, as they have never seen it as under threat to begin with. Thus the association of the will to live with existentialism: the existentialists accept that there is a legitimate challenge to be made, but believe it can be overcome. There are three basic ways of responding, which might be referred to most generally as theistic existentialism, atheistic existentialism, and agnostic existentialism.
Theistic existentialists accept that the traditional arguments for God fail (as even Kant had admitted), and that the pessimists’ lack of belief in God is understandable if rationality is the only way of approaching belief in God. They reject the view that rationality is the only way of approaching God, however, and emphasize the fact that faith has always been an essential element of belief in God. Faith is non-rational, but they insist that it is not irrational. Through faith, we come to have a real belief in God rather than the weak belief susceptible to changes in philosophical and scientific knowledge. This belief also restores God’s ability to be a foundation for our lives and morals. Thus we construct a meaningful life for ourselves in relation to God and have faith that it mirrors the meaning God has for us and the world.
Atheistic existentialists accept that the traditional arguments for God fail and do not seek to replace them with alternative justifications for God. Thus they abandon the possibility of life having inherent meaning or value that must be approached non-rationally. Instead, they take it that we construct a meaningful life for ourselves by creating meaning for ourselves and without relying on God to justify it in the end. They take the pursuits of theistic existentialists to themselves be meaningless because there is no inherent meaning or value in the world for our constructed meanings to reflect. Yet they also take it that this mirroring relationship is unnecessary. Life only has negative value if you fail to value it for what it is, they argue, and thus the affirmation of life as it is—and not as we once wished it to be—is what they take to be essential for finding one’s own way of living a meaningful life.
Agnostic existentialists, more commonly referred to as absurdists, suspend judgment with regard to the questions that divide theistic and atheistic existentialists. Perhaps there is an inherent meaning to the universe that we can reflect in our lives, but that it cannot be rationally approached or appreciated means that it is necessarily beyond human grasp. What we must do, then, is reject attempts to hide the problem of the absurd from our mind. Attempting to rely on God is a subtle rejection of this life, the absurdist argues, for it implicitly accepts that the pessimist is correct unless something that we cannot possibly know turns out to be true. Yet attempting to create our way of living can also be a way of hiding the problem of the absurd if it is used as a means of distracting ourselves from the fact that life has no inherent meaning or value rather than as a means of confronting that fact.
This should give you a much more thorough picture of the philosophical landscape here than the simple dictionary definition. When an existentialist talks about a free and responsible person, then, he is not talking about a person with responsibilities in the sense of an employee or a parent. He is talking about a person who recognizes himself as part of the great causal chain that determines facts about himself and the world around him. Such a person realizes that he is not merely acted upon by the world, but in fact acts upon the world—which includes himself. We obtain the will to live, then, by not giving in to the pessimists’ inability to see how life could be worth living in the absence of some external factor telling us how to live. In the end, it is the pessimist who is revealed as an infant who has never learned to walk—a desperate follower without a leader.