Backend: Remove the previously used `include:with` syntax

Summary

As a part of the proposal discussion in #406780 (closed), it was proposed to change from using include:with to include:inputs which would allow for future expansion.

Other benefits:

• This is still an industry term, so there should be familiarity with it.
• It'll be easier for newer people to intuit the relationship between the component and your own pipeline. "This component requires these inputs, so I'll add the inputs to the includes block".

So these would work together:

• spec:inputs: Used in the component header, to define the inputs a component needs.
• include:inputs: Used in the pipeline, to set the input values for the included component.

Proposal

Once include:inputs is introduced in #406780 (closed), this issue will be to remove the previously used include:with syntax.

NOTE: Make sure the docs are also updated (not sure if we have this or not)