Fox or Tanaka: which max heart rate formula should I use?
There is no closed answer using a formula alone: Tanaka (2001) was published as an update to Fox (1971), and the gap between the two grows with age, from 0 bpm at 40 to 6 bpm at 60. An exact max heart rate value only comes from a direct physical test, and choosing between estimates should involve guidance from an exercise or health professional.
What is the Karvonen method and why does it need resting heart rate?
Karvonen calculates the training zone from heart rate reserve, the gap between maximum and resting rate, instead of applying the percentage directly to the maximum; that is why it needs a measured resting rate, a piece of data no age-based max heart rate formula supplies on its own.
Why does the vigorous zone shift so much between the simple method and Karvonen?
Because the simple method applies the percentage directly to maximum heart rate (for example, 70% of 180 bpm), while Karvonen applies the percentage to heart rate reserve and then adds resting heart rate back in; the lower a person resting rate, the larger the reserve, and the bigger the gap between the two results.
What does training in the "moderate" heart rate zone mean?
It is the range between 50% and 70% of estimated maximum heart rate, the percentage definition used by this calculator; it is an intensity classification, not a training prescription, and any directed exercise plan should account for individual health history with professional support.