There are a lot of different names for arguments for God's existence.
There are some unique arguments, like the argument from desire. There are also some argument families (by which I don't mean the people you invite for Thanksgiving) that, in my experience, tend to be called by one name, such as the ontological arguments.
I find the arguments from design and contingency to be the most confusing, because there are more variations to these arguments and many ways to name them.
Arguments from contingency focus on some thing A that relies on some thing B. Those things might be motion, existence, goodness, and so on. St. Thomas Aquinas put forth four of these arguments and people ever since have been naming them different things. (Yet I'm still sticking to it all being Aquinas' fault.)
We can talk about the first efficient cause of the beginning of the universe,
or the first efficient cause of the beginning of the universe,
or the first efficient cause of the beginning of the universe.
Ultimately, we're making the same (or a very similar) argument, but we're using a different name and, perhaps, emphasizing a different aspect of it. You say "motion"; I say "movement". Tomato, Thomato.
The point is that there are many arguments, but if we understand the broad idea of arguing from contingency, we're well on the way to understanding any particular argument from contingency. As well, we shouldn't let all the terminology confuse us--at least no longer than it has to.