Does bivariate normal distribution mean the two random variables have normal distributions? is that enough for two random variables to have a bivariate normal distribution or are there some other conditions that must be true?
A bivariate Normal distribution is a special cause of the multivariate Normal distribution in which the dimension is 2. It describes a single random variable, not 2 random variables
However, 2 independent univariate Normal random variables can be expressed as a special case of the bivariate Normal random variable with a diagonal covariance matrix (zeros of the off-diagonal).
Have a look at the second figure in this article for a visual explanation of the bivariate Normal distribution and its two corresponding marginal distributions which are univariate Normal.