I think you're interpretation is correct and GP is wrong to think the "primary driver of the Fed is stopping prices from increasing too fast."
Fed has recently repeatedly stated it's goal is to drive down unemployment and is willing to let inflation run hot to achieve that. It's always a balancing act if one of them is worse than the other, they will focus on that. However it looks like when both of them are bad, they would rather focus on unemployment by letting inflation hit harder.
The Fed has a formal target of 2% inflation (the major change recently was to allow it to be 2% on average, over time rather than 2% period)
It doesn't have any employment target, although obviously extremes of unemployment is something it pays some attention to when deciding whether it's OK to miss its primary target or not.
