It seems you are both correct, and the disagreement lies on the meaning 'personal':
- They use 'personal' as 'private', or 'information about you'. e.g. I have a fetish, I wish no one to know, it's 'personal'.
- You seem to use 'personal' as 'personally identifiable information'. e.g. if I have this data, I can know trace back to its originator.
The ambiguity can be found in many places around. Most people don't make the distinction. Rather, they think anything private is personal. At the same time, the definition you use can be found in official documents, like GDPR, the new EU Privacy Law (https://gdpr-info.eu/art-4-gdpr/).