Can a Homogeneous Differential Equation be Nonlinear? First of all, I am not asking about $v=y/x$ transformation kinda homogeneous. 
Can we say  this nonlinear differential equation is homogeneous $$y^{\prime}=ty^2.$$
Here there is no term without $y$, this is okay. But this is nonlinear equation.
I saw here some discussions and some people said it is just for linear equations. I want to give two links:
Dummies Series (assumes it can be nonlinear) (I also see some universities documents too).
Wolfram Math World (assumes it has to be linear)
Which one is the definition:
1) No term without $y$,
2) Linear and No term without $y$.
3) $y$ is a solution, then $\lambda y$ is solution for all $\lambda \in \mathbb R$.
Considering the references, I will use the 3rd one. This option is also meaningful.
 A: The answer to the title-question is: Yes, it can. Here are two references.

The paper On the second order homogeneous quadratic differential equation by Roger Chalkley considers homogeneous quadratic differential equations of the form
  \begin{align*}
Q(y)\equiv a\left(y^{\prime\prime}\right)^2+by^{\prime\prime}y^{\prime}+cy^{\prime\prime}y+d\left(y^{\prime}\right)^2
+ey^\prime y+fy^2=0\tag{1}
\end{align*}
from which we conclude linearity and homogeneity are two  concepts which do not exclude each other.

Note: The Math world page does not include the term linear in the definition of Homogeneous Ordinary Differential Equation. It rather uses a linear differential equation as an easy to follow example for a homogeneous ordinary differential equation.

Another example is the paper
Classification and Analysis of Two-Dimensional Real Homogeneous Quadratic Differential Equation Systems by Tsutomu Date.

A: It is to some extent a matter of convention and as with many things different conventions exist.
Consider the following related notion: a function $f : \mathbb R \to \mathbb R$ is called homogeneous of degree $\alpha$, if $f(kx) = k^\alpha f(x)$. We could apply this to ODEs and say that an ODE 
$$
y'(t) = f(t, y(t))
$$
is homogeneous, if the right hand side is homogeneous in the $y$-variable. With this definition your example would be homogeneous of degree 2.
I would not call the ODE
$$ y'''' + ty'' + y^2 = 0$$
in the first example you linked homogeneous. Informally, the term homogeneous implies that we have some form of scaling behaviour.
I have come across homogeneous ODEs only in the context of linear equations, but it is possible that more general notions exist.
