Q: this did not work on either white 2010 Macbook or 2010 mabook pro running  El Capitan and Yosemite respectively  is there something special for these releases?

For model builds before 2013 the diagnostic test uses a different method.

Two different support documents cover most aspects; there are reasons

why the AHT vs Diagnostics may fail to work in certain build year models.

And with specific systems, the Recovery partition (or lack thereof) matters.

• Using Apple Diagnostics - Apple Support

For computers that shipped with OS X 10.7 or earlier, the test is on the

included system disc media kit; so you'd start from there to run test.

• Using Apple Hardware Test - Apple Support

• Finding Hardware Faults: Exploring AHT & Apple Diagnostics

Before You Run AHT/Apple Diagnostics. By controlling both the software and

hardware distribution for its systems, Apple is able to optimize the two halves to work ...

http://computers.tutsplus.com/tutorials/finding-hardware-faults-exploring-aht-ap ple-diagnostics--mac-60456

There are a few ideas and methods in the articles above.

A reason for the issue you face, is the vintage of hardware & macOS they shipped with.

Good luck & happy computing!

AHT is on disc 2 of the gray install discs that shipped with the computers. Each model has specific discs so one won't work with the other.