Say I have a matrix,
$$\left( \begin{array}{cc} a & d \\ b & e \\ c & f \end{array}\right)$$
I am confused when people say columns, do they mean only the $$\begin{array}{c} a \\ b \\ c \end{array} $$ in the matrix or this vector $$\left( \begin{array}{c} a \\ b \\ c \end{array} \right)$$ like do they mean the entries of column in the matrix or a column matrix with same entries as that in column of the given matrix.
Some lecturers seem to use it interchangeably. I'm not sure if I'm even making sense but for instance, $AX=b$ is the system of equations, where X is a column vector then $AX$ is the linear combinations of columns of A? Wouldn't it be more correct to say, column matrices instead? Or is column a matrix in its own right? Well that can't be right because then I have more questions ...
A column matrix is an ordered list of numbers written in a column. However, you are right that column and column matrix are terms which are used interchangeably.
In practice, what is meant will be clear from the context. So, in your example, your lecturer clearly meant that $AX$ is a linear combination of the column matrices $\left( \begin{array}{c} a \\ b \\ c \end{array} \right)$ and $\left( \begin{array}{c} d \\ e \\ f \end{array} \right)$.
Ideally, you should use the terms precisely but don't expect everyone else to do the same!