Lambda Calculus Argument Delimiter
So, I think part of the problem is that the resource you're using is not using standard lambda calculus notation.

In most texts, there is a space between arguments. I would write $\lambda x\ y\ z\ldotp z$ in place of $\lambda x y z\ldotp z$. But, if you assume a convention that variables are always single letters, you can omit the space without ambiguity.

But I can't imagine one would ever write whole programs this way. Rather, this is more likely to show up in examples, code snippets, and small proofs, where there aren't many variables and the names don't matter.

Basically it boils down to this:


A minimal lambda calculus only allows one argument per function, so there's no need for a delimiter. The dot denotes the function body.
There is common syntactic sugar for multiple arguments, which abbreviates $\lambda x \ldotp (\lambda y \ldotp (\lambda z\ldotp z))$ as $\lambda x\ y\ z\ldotp z$.
In a situation where variable names are assumed to be single letters, one could use $\lambda x y z\ldotp z$ in place of $\lambda x\ y\ z\ldotp z$. 


There is no delimiter, other than the space, that I know of for arguments. Commas would be frowned upon, since they indicate tuples, rather than curried arguments.

EDIT: to respond to the OPs edit:

There is no such thing as multiple arguments in an untyped lambda calculus, at least not in standard ones. There are two ways to simulate multiple-argument functions.


Tuples: you can use a Church encoding to pass a single argument, which can be decomposed into multiple values. So you would write $f (&lt;x,y&gt;)$, where $&lt;x,y&gt;$ is some special syntax for making a pair, and then decompose it as $HD\ t$ and $TL\ t$ inside your function to get at each argument.
Currying: When, in an imperative language, you would write f(x,y,z), you write $(f\ x\ y\ z)$, which is syntactic sugar for $(((f\ x)\ y)\ z)$. A function accepting multiple arguments is just a function which returns another function.


It turns out that these two are basically isomorphic, so which you do doesn't much matter.

Some languages, like Scheme, have built in syntax for multiple-argument functions, but I haven't seen anything like that in a formal lambda calculus. Usually they're intended to be minimal, and so when you don't need multiple-argument functions, you don't put them in.