What is an "origin pole"?
The "origin pole" is indeed the \$1/s\$ term in the transfer function \$H(s)\$. In the bode plot it results in a first order transfer that does NOT flatten out for low frequencies.

Your Bode plot is that of a low pass filter $$H(s) = \frac{1}{1 + s}$$
Note how this \$H(s)\$ would result in \$H(0) = 1 = 0\text{ dB}\$ like in your Bode plot.

\$H(s) = 1/s\$ is different, \$H(0) = \infty\$! In theory at least.
So the -20 dB /decade line in the Bode plot keeps going on forever to both sides. Note that a Bode plot has a logarithmic X-axis, where would that place the 0 Hz point ? At minus infinity!

I call this \$1/s\$ an integrator or pole at zero, they are useful in feedback loops to eliminate static errors. Almost every PLL has an integrator consisting of a charge-pump (switched current source) feeding current into a capacitor. What happens to the capacitor's voltage when you feed a current into it? Yes, it keeps rising forever. That's integrator behavior.