I have two related questions about terminology.
If a matrix contains probabilities such that each column (or row or both) sums to $1$ , is this matrix always called a stochastic matrix i.e. even if the probabilities don't represent transition probabilities of a Markov chain? If not, does this matrix have a name?
What is the name, if there is one, to the matrix that contains probabilities such that the sum of all the elements is $1$ (i.e. the elements represent joint probabilities)?
A stochastic matrix is, by definition, any matrix for which each row, column or both, sums to $1$.
Mathematics does not concern itself with what mathematical objects "represent". It only cares about what their properties are. And what you described is a stochastic matrix.
In fact, every stochastic matrix will be a transition matrix for some Markov chain, so, in a way, your theoretical matrix which has row sums of $1$, but is not a transition matrix, does not even exist.
For $2$, I know of no short name for such a matrix.