As this article notes, Clairol products have eliminated the repeat instruction while others use terminology like “repeat if necessary” or “repeat as desired”. I can remember the days of once or twice a week shampooing in the mid sixties where my hair was also tortured into once a week pin curls with lavish amounts of Dippity Do. Mom burned through the Aqua Net getting that updo. Men were still using that greasy kid stuff. It took a lot of Prell to get that stuff out of your hair.
Honestly you wont. Once you let your hair recover, it will be fine. Yes, for a bit it will look and feel odd.
So you understand (correctly, I think) that the first wash really DOES do something – it strips away the oil. The failure to lather is nothing more than the evidence that the shampoo is chemically reacting with the oil, the oil molecules glomming onto the oil-philic parts of the soap molecules (creating the basically non-lathering soap scum) which you then rinse away.
The second wash, with is full luxurious lather, simply demonstrates that the first wash did its job (at least mostly anyway).
ETA: Okay, I just did my google homework: The word I’m looking for, for “oil-philic” is oleophilic or possibly lipophilic.