I don't think that diagram proves what you think it proves...You are mistaken. Many BMSes intended for use in high current applications have specific port(s) for connecting charge/discharge contactors. Here is a wiring diagram for Zeva BMS16, with two separate contactors clearly visible :
http://www.zeva.com.au/Products/images_lge/BMS16v2_Example.jpg
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It's all good.Except you said
Which I explained it often can't do, as there there may be no integration between the BMS and the charger.
I also responded to jbman thinking that was your comment. Sorry jbman![]()