The Moderna and Pfizer Covid vaccines use mRNA vaccine technology.
The Oxford-AstraZeneca Covid vaccine uses viral vector vaccine technology. It was not approved or authorized for use in the US.
The earliest reports of symptoms of transverse myelitis were by British neurologist Henry Bastian in 1882 and 1910. The condition has been studied to discern it from diseases such as MS, with defining diagnostic criteria being established in the 1980s. A drummer for metal band Slipknot was hospitalized with transverse myelitis in 2013. The first vaccines for Covid started clinical trials in early 2020.
It has been associated with both bacterial and viral infections. Based on the OP article and the association with infections, it appears that transverse myelitis at least sometimes (apparently ~60% of cases' causes are unknown) happens when the immune system is activated, which happens with infections and with vaccines. More simply, it's an auto-immune condition that happens when the immune system is activated.