Which Teams to remove?

Yes, you're correct that "Classic" Teams is nearing the end of life, and it's best to move forward with the "new" Teams, which is the updated version. The "new" Teams is the modern experience that Microsoft is focusing on, and eventually, Classic Teams will be phased out.

Regarding the update dates, it’s possible that "Classic" Teams received an update to prepare for its transition or to maintain compatibility for the remaining users, while the "new" Teams might not show an update for the same reason—it might already be up-to-date with the latest features.

Since you work with both private and changing business accounts, you’ll want to stick with the new Teams because it’s more flexible and features updates that make it easier to handle different types of accounts, especially when collaborating with multiple clients. Also, the new Teams is designed to better handle the evolving work environments, including features like Teams for personal use, easier management for hybrid work, and improved integration with Microsoft 365.

If you're still unsure about which version you're using, you could check the Teams version in the settings under About in the Help menu to confirm which is the "new" Teams and which is "Classic." The "new" version will also be labeled as "Teams" and should be the one you keep.