Once we have defines the angular momentum operators $L\_z,L\_y,L\_x,L^2$ ($L^2=L\_z^2+L\_y^2+L\_x^2$) suppose we focus on the eigenstates $|l \ m \rangle$ common to both $L\_z$ and $L^2$:
$$L\_z|l \ m\rangle =\hbar m |l \ m\rangle$$
$$L^2|l \ m\rangle =\hbar ^2 l(l+1) |l \ m\rangle$$
suppose now that we want to talk about eigenfunctions of the angular momentum operator instead of the eigenstates of it, every source that I could find performs this switch in the following way:
$$\langle \theta \ \phi |L\_z|l \ m\rangle =\hbar m \langle \theta \ \phi |l \ m\rangle \ \Rightarrow \ -i\hbar \frac{\partial}{\partial \phi} Y\_{l,m}(\theta,\phi) =\hbar m Y\_{l,m}(\theta,\phi) \ \ \ \ \ \ (1)$$
My question is: why are we putting ourselves in the base of $\theta \ \phi$ instead of $\theta \ \phi \ r$? In other words why do we have equation (1) and not the following:
$$\langle \theta \ \phi \ r |L\_z|l \ m\rangle =\hbar m \langle \theta \ \phi \ r |l \ m\rangle \ \Rightarrow \ -i\hbar \frac{\partial}{\partial \phi} Y\_{l,m}(\theta,\phi , r) =\hbar m Y\_{l , m}(\theta,\phi,r)$$
Since we are working in 3D space I would expect the wave function $Y\_{l,m}$ to represent the probability amplitude in 3D space; I would like $Y\_{l,m}$ to be something that I can square and then integrate over 3D space to get a probability, right? Seems strange to switch from cartesian to spherical coordinates and then ignore one of the spherical coordinates..  

In every lecture I could find everybody uses only $\theta$ and $\phi$ without $r$ but then fails to explain why.

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P.S.  

I have noticed that $Y\_{l,m}$ are usually called "spherical harmonics", I'm not sure that this is relevant here but for now I have failed to understand the reason behind this name, so maybe this has something to do with this other thing I don't understand..