Ok,

the first effect generally means for put option, the value of the option tend to rise if it has a longer time to maturity (more insurance value).

However,

For put option, one can sell the stock and receive the cash K. One may tend to get this K as soon as possible so that he will gain more time value of money (e.g. deposit K to get more interest). This second effect makes the put option's value can be lower as the time to maturity goes up. For deep in the money put, this effect is so huge that make the time value of the option to become negative. So this is why the american put may be early exercised.

Best,

Do Bermuda put options have the same second effect as American put options，or not？

yes, I think so