No, not from Earth.
According to Elon Musk's most recent comments at SpaceX's 2019 Starship Update presentation, Starship cannot reach orbit without the Super Heavy first stage booster, at least on Earth. (Though when launching from the moon or Mars it's no problem.)
Previously in 2018 Musk had stated on Twitter that Starship would be technically capable of SSTO, but only with no payload (making it entirely pointless to try to use Starship that way). Later in May 2019 he reiterated that Starship could do SSTO, but only with the heat shield, landing propellant, and legs stripped off. (Meaning no way to get back to Earth after launch.)
With these older comments in mind, it's not entirely clear whether Musk's most recent statement at the Starship Update Presentation means the latest Starship design can't support SSTO at all (even in a hypothetical expendable configuration with no useful payload) or if he was only referring to SSTO with a useful payload, particularly since the question he was responding to included that qualifier. In either case, the distinction between "unable to do SSTO" and "unable to do SSTO with a non-zero payload" is mostly academic, since in normal operation Starship won't ever be launching from Earth without Super Heavy.
As for the Space.com article, I can see how that wording might be confusing, but I don't believe they're actually trying to suggest that Starship will be doing orbital tests without Super Heavy. Rather, "the first operational Starship-Super Heavy flight" refers to the first flight that will actually be carrying useful payloads to orbit; it doesn't count the test articles and prototypes they're using for development.