What are the differences between multivariate data visualizations and multidimensional visualizations?
I don't think there is a well-defined (or well enough defined) terminology for that. 

Multi-variate means that it involves multiple (dependent or independent) variables. Which is pretty clear.

Multi-dimensional in mathematical sense means that the data points are from a multi-dimensional space (typically vector space). This is exactly the same as multi-variate. In this sense, your visualization is both multi-variate and multi-dimensional. And both the size of the markers and the color represent individual dimensions or variables, which is then equivalent. Also, time would be an individual dimension. Personally, I would go with this interpretation of the term multi-dimensional.

If you want to restrict the term multi-dimensional to spatial dimensions of the plot, then size and color of markers would not be dimensions in that sense. I personally find this interpretation confusing. Time then also would not be a dimension in this sense, unless you represent it by a 3rd axis in the graph. This, again, is confusing as many people actually often put time into the same category as spatial dimensions.

So, all in all I would stick to the mathematical interpretation of dimensions here so that multi-variate and multi-dimensional will be equivalent.