Brown Recluse can also be devastating as well. The venom from a brown recluse can cause severe necrosis in the skin.
I believe in Australia though, they only have the Redback Spider which is a cousin of the Black Widow.
You are absolutely right that there are differences between the Australian black widow and the North American black widow. However, black widows are found not only in North America, but also in Southern Europe (Mediterranean black widow), China (Latrodectus elegans) and South America.
Scientific publications classify the redback spider as a modality of black widow. And "Australian black widow" can be used as a synonym for redback spider.
I found these academic papers which employ the terms "redback spider" and "Australian black widow" as synonyms.
"redback Australian black widow spider"
Pires, O. R., Fontes, W., & Castro, M. S. (2016). Recent Insights in Latrodectus (“Black Widow” Spider) Envenomation: Toxins and Their Mechanisms of Action. Spider Venoms, 333–344. doi:10.1007/978-94-007-6389-0_23
"Black widows are cosmopolitan and found throughout much of the world: China, Central Asia (L. mactans tredecimguttatus and L. lugubris, or karakurt), southern Europe (L. mactans tredecimguttatus), North and South America (L. mactans, L. geometricus, L. hesperus), India and Australia (L. hasselti, or red-back spider)"
Ushkaryov, Y. A., Volynski, K. E., & Ashton, A. C. (2004). The multiple actions of black widow spider toxins and their selective use in neurosecretion studies. Toxicon, 43(5), 527–542. doi:10.1016/j.toxicon.2004.02.008
"The redback spider or Australian black widow"
Nentwig, W., Mebs, D., & Vilà, M. (2017). Impact of Non-native Animals and Plants on Human Health. Impact of Biological Invasions on Ecosystem Services, 277–293. doi:10.1007/978-3-319-45121-3_18