Macbook pro showing "Starting Internet Recovery" instead of AHT

From Apple's documentation on AHT:

Some Macintosh computers that shipped with OS X Lion and later support the use of Apple Hardware Test over the Internet. These computers will start up to an Internet-based version of AHT if the hard drive does not contain AHT. An Internet-enabled connection via Ethernet or Wi-Fi is required to use this feature. Internet-based AHT functions the same as AHT on the hard drive or flash storage outlined above.

What you are using is AHT, it just doesn't have it pre-installed on your hard drive.

EDIT: You should have the option to perform extended tests, it will be a checkbox you have the option of selecting before running the tests

EDIT 2: As your mac was made after 2013, you are forced to use Apple Diagnostics, which is why I assume you can't have access to the extended test options you see on the AHT documentation.