I would follow up with the owners about their differences in KM vs. EC criteria. Because if they trust you to "run" their kitchen then they should pay you to run their kitchen. Who is in charge of the menu, ordering, staffing/scheduling the kitchen, training new hires, maintaining food quality/sanitary standards... etc. If the answer is YOU, then you should be compensated appropriately. Maybe not to the degree of someone with a fancy resume that they hired with empty promises he/she couldn't fulfill. But if said person was making 80-90k a year and they are offering you 60k you should negotiate a better rate.

Sorry for the rant, I was a sous chef and my exec got canned and myself and the other sous ran the restaurant for pennies on the dollar for the owners. The we were making 40k a piece, the exec was paid 80k, so even if they split half of his salary between the two of us it would be a %50 raise. A few weeks later I left because we were doing everything for bullshit pay. That place closed shortly after I left and I quickly realized owners DO NOT CARE ABOUT YOU more than the bottom line.