The article isn't about how many types - or rather cardinalities (sizes) - of infinity exist, it's about which of those cardinalities describes the real numbers

The article isn't about how many types - or rather cardinalities (sizes) - of infinity exist, it's about which of those cardinalities describes the real numbers. "Countably" infinite sets have cardinality Aleph_0, the "smallest infinite size". There is an infinity of "larger infinite sizes", all of which are "uncountable", so to refer to a set as "uncountably infinite" doesn't pin down which particular cardinality it has. The article is about what specific cardinality the set of real numbers has, and in particular whether it's Aleph_1 or Aleph_2 as it seems less likely to be any of the infinite other possibilities.