Confusing Lines About Extinct Elements
Defintions

Occur on Earth differentiates between all isotopes in the universe and isotopes which exist or have existed on Earth. The passage identifies that there are a total of 308 isotopes occurring on Earth, but it is worth noting that there are more than 308 total isotopes in the universe. This discrepancy is because some isotopes have extremely short half-lives, making their total "lifetimes" extremely short. As an example, an isotope generated in a supernova which has a half-life of 1 second will have totally degraded in at most 32 seconds (32 half-lives degrades the sample completely no matter the original amount). That is not nearly enough time for the isotope to travel from the exploded star to Earth, so that isotope does not occur on Earth (and it never can). This hypothetical supernova isotope would exist but it would not be part of the 308 Earth occurring isotopes.

Extinct means that the isotope is no longer naturally occurring within a given environment. Isotopes have differing half-lives, and a currently extinct isotope would have degraded so completely as to no longer be present at all. The isotopes would have originally been seeded into the environment by processes like supernovas, which produce ejecta containing the elements produced by nuclear fission and nuclear fusion. When that ejecta reaches the Earth the isotopes within it begin to occur on Earth, but eventually those isotopes will disappear due to half-life degradation. There are processes which can replace degraded isotopes in a self-regenerating manner and thus those isotopes never go extinct due to self-regeneration. An example of this type of self-regenerating isotope is Carbon-14; in fact, Carbon-14 will likely never become extinct. Self-regeneration is not the only process to replenish isotopes; the aforementioned supernovas are another. Unlike self-regeneration, which is a continuous process in which degraded isotopes continuously get replaced, supernovas deposit isotopes intermittently and comparatively infrequently. In fact, if a supernova deposit is delayed for an extended amount of time, isotopes which come only from supernovas can actually disappear from Earth entirely- they would become extinct. These extinct isotopes would stay extinct on Earth until a supernova redeposited the isotope.

Clarifying the Passage

As far as your question about an extinct isotope occurring on Earth- for the isotope to be extinct it cannot be currently occurring on Earth, but it was present in the past (logically, if an isotope never appeared on Earth it couldn't be part of the 308 isotope total, so for some of that 308 total to be extinct means they were present on Earth in the past). That question is sort of unrelated to the passage however, but the passage was written poorly enough that I think you misunderstood that they were trying to say that 121 isotopes should hypothetically be extinct, but that some of them are not actually extinct. To reduce the confusion the passage is causing, I have rewritten it to be far more clear: "Among all the elements that have have ever occurred or currently occur on Earth there are 308 total isotopes. Those isotopes are split into 150 stable isotopes and 158 unstable isotopes. Of the 158 unstable isotopes, 121 should be or currently are extinct, however, there are isotopes not actually extinct because they are constantly renewed, like carbon-14."