How to detect underdamped harmonic signal?
First, de-noise the waveform using a technique suitable for non-stationary signals. Admittedly, the spectrum of the signal doesn't change too much over time. There are efficient iterative de-noising algorithms for this purpose, such as basis pursuit de-noising [1]. But even a simple low-pass filter may be sufficient depending on how bad the non-stationarity is.
Since your record is mostly oscillatory, the underdamped signal effectively begins when the successive peaks of your signal are lower in amplitude. With your de-noised smoothed signal, attempt to pick up local extrema. This can be done in real-time by looking at the first derivative or discrete difference. Once you have extracted the extrema, using the values of the extrema, you can construct the signal envelope. Empirically, you can decide that if the envelope is decaying, then it signifies the starting point for under-damped behavior. 


Fourier and short-time Fourier transforms may not be as effective in this problem since you are looking for the exponential decay feature, which is a temporal signature.

[1] Chen, Scott Shaobing, David L. Donoho, and Michael A. Saunders. "Atomic decomposition by basis pursuit." SIAM journal on scientific computing 20.1 (1998): 33-61.