Do I need to extend my Thailand tourist visa if it's valid for 90 days but only allows a 60-day stay?

The visa is used. Single entry.

You have a single entry tourist visa that is valid for 90 days, between Jan to April. That means you can enter Thailand ONCE (anytime you want, between the issue date and the expiry date) and upon entry you will receive a permission of stay of 60 days, that you can further extend at local immigration for 30 days for 1,900 baht.

Example: Your visa is valid until April 5th. If let's say you enter on April 4th, you will receive a permission of stay until June 3rd, that you can further extend to July 2nd at local Immigration.

You have to extend. The visa is valid for 90 but your stay is max 60 days, then extend.