There isn't a particular name for someone between in 50-60 but someone over 60 is usually called an OAP (Old Aged Person).
A person aged between 50 and 60 years is often referred to as "middle-aged."
A person aged between 60 and 70 years is typically referred to as a senior citizen or someone in their senior years.
A person Aged 90 to 99 years called a Nonagenarian.
aged people
Senior citizen.
old
A person aged 100 years is commonly referred to as a centenarian.
An octogenarian.
A Nonagenarian
Gax
tricenarian