alt Hacker NewsI actually work in this field, so I can encourage you to do some further reading and get to grips with the terminology. "Indo-European" remains among experts the most popular way to refer to the entire language family, Anatolian and non-Anatolian. Terms like "Nuclear IE" and "Core IE" are often employed to distinguish the non-Anatolian languages.
"Indo-Anatolian" for the higher-level family is a term that some may use here and there, but no expert would ever make a "well, acktually..." remark as you just did to those in a discussion, professional or layman, using "Indo-European"
It's not consensus (yet?), but Kloekhorst's categorization does put the Anatolian split first and early, making it essentially two separate branches of PIE. He isn't concerned about the terminology, but it does allow for alternatives, and "Proto-Indo-Hittite" is now a century-old term.
Despite the terms used, Anatolian and non-Anatolian languages are seen as two separate branches.
Maybe I'm being too charitable, but I hadn't read that user's comment as a "well, acktually", especially given the chain of comments, but perhaps I'm wrong.