"....that total strain is the total engineering strain or total true strain?" - Its total true strain, as mentioned by Bhargava at 4:27.
"So why should the elastic strain ( = true stress / E) be not equal to the total strain?" - They should be equal. True stain = ln (1 + engineering strain). How much of a difference are you getting between the LHS and RHS? For steel with yield strain = 0.002, this difference is 0.1%.
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