ARMA(p,q) is generally denoted as a special case of
d = 0. However,
ARIMA(p,d,q) is actually
ARMA(p+d,q) so an ARIMA is actually an ARMA model, right? Then, how come ARIMA model is generalized version of ARMA models?
If you do not impose any restrictions on the coefficients then yes, the general ARMA model is the most general, and it subsumes the ARIMA model. The general ARMA model includes both the ARIMA model as well as "explosive" cases. However, it is common to impose the implicit condition that the auto-regressive part of the ARMA model is stationary (autoregressive roots outside the unit circle), and in this latter case, the ARIMA model (with stationary AR part) is the more general, and it subsumes the stationary ARMA model. The relevant subset relations are shown in the diagram below.
This issue can be rather confusing because time-series books and notes ofen fail to explicitly differentiate between the general ARMA model (allowing any coefficient values) and the stationary ARMA model (requiring the roots of the AR part to be outside the unit circle). Many texts regard the "explosive" cases as being of no interest, and they also wish to exclude the ARIMA model from the ARMA form, so they will often implicitly assume they are dealing with the stationary ARMA model without mentioning the constraint explicitly.