What does Apple.com/support -3403d mean when trying to run diagnostics check on my MacBook Pro?

what model MacBook Pro?

older Macs, before about June 2013 do not use "Apple Diagnostics" so when you try to get the diagnostic via Internet Recovery, it cannot be found and you get an error.

How to use Apple Diagnostics on your Mac - Apple Support

On your older Mac, the Apple Hardware Test will be found on the original-Installer DVD. If you need help finding a copy on the Internet, please post back.

In Internet Recovery, these low negative thousands error messages indicate that your provided Wi-Fi Network is inadequate in some fashion, and cannot be used (for this purpose) in its current state.

Common problems include:

Hidden Network-name

required use of a proxy server to get Internet access

required use of a login page to get Internet access

Use of a login server or certificate to get Internet access

Use of PPPoE in the Mac to get Internet access (typically only applies to DSL)

Use of Fixed IP address rather than good old DHCP

Variance of the time by more than five minutes

Failure of Router to provide workable DNS server addresses, or providing 0.0.0.0 [may produce -2002f]

If you have an Ethernet port, you may in some cases be able to move your Mac close to the Router and connect to the Router using Ethernet.

The higher the number, the more obscure the problem.

Also, make certain the Date&Time and Time Zone are correct within 5 minutes.

A User collected the ones Apple left out in the open and put them on a GitHub web site, along with instructions for making a stand-alone USB-stick (execute-only). Installation back in "the right place" on your boot drive is very complex, and too much trouble.

https://github.com/upekkha/AppleHardwareTest

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Do you see an image of a globe with a triangle containing an exclamation point?

That means it's trying to either access MacOS Recovery or Apple Diagnostics through the internet and can't connect. That would be a sign that the tool you're trying to access is not present or is damaged on your drive.