While not having read it closely myself, I remember that there is an appendix in [this](http://www.kurims.kyoto-u.ac.jp/~motizuki/Topics%20in%20Absolute%20Anabelian%20Geometry%20I.pdf) paper
by Shinichi Mochizuki claiming to develop the theory of Albanese varieties in modern language. Maybe you find it helpful.
Skimming it, it seems that his main result is, that any separated geometrically integral scheme of finite type over a field admits an Albanese.