The Wikipedia article is more technically correct, in that the term RNN is formally taken to mean "a neural network with recurrent connections", and that includes many architectures that match this description, including LSTMs.
However, it is also common to see "RNN" used as a short-hand for a kind of "Vanilla RNN" or "basic RNN", where one or more layers have weights connecting the layer to itself (its own activations from $t-1$ are concatenated to the external inputs at $t$), and there are no other gates or special combinations, just those recurrent connections.
Oddly, this basic layer-based RNN archtecture is not listed in all the options on the Wikipedia page on RNNs - probably the closest are Elman networks and Jordan networks which are ways to implement the recurrent connection. It is a valid architecture choice, and can be effective. The LSTM and GRU architectures improve on it in terms of handling longer sequences and preserving important signals over them when training (e.g. matching a starting and ending quote in text processing).