I have just remembered something that Simon Kinberg said back in 2017 to Entertainment Weekly where he told them the premise of the film. Described what was happening to Jean:
“Jean absorbs those solar flares and then hurls them back out into space,” explains director Simon Kinberg of the sequence shown in the above exclusive concept art. “That’s what triggers, awakens, creates this Phoenix force and voice inside her. What appears to the world to be another heroic victory by the X-Men in saving stranded people in space we realize is the the thing that begins the unraveling of Jean and the unraveling of secrets Charles told her.”
So the cosmic energy isn’t meant to be the Phoenix Force but the catalyst for it. Vuk said it was “drawn” to Jean, saying that it is drawn to Jean because is stronger than she knows and is “special”. Suggesting that the Phoenix Force is Jean’s power at its peak.