How does an "Agile SDLC" fit in with the concept on an "Agile Methodology"?

Does the SDLC fall under the team's methodology?

Would it improve empiricism in value delivery and the ability to innovate and learn?

Agile, itself, is a mindset

I'm not sure that I'd fully agree with that statement. Agile is a set of values and principles. For software development, those values and principles are defined in the Manifesto for Agile Software Development. Similar values and principles have been adopted by other domains.

An agile framework, is a predefined structure, such as scrum. Frameworks as usually a starting point for a teams agile methodology.

Overall, I think this is a fair assessment. Different frameworks may offer different amounts of structure or structure over different aspects of the development life cycle.

An agile methodology, is a set of convention that a team agrees to follow, that align with the agile values

A methodology is more than a set of conventions. I don't see a real distinction between "methodology" and "SDLC". Both describe the processes and activities that the team (or teams, in a scaled context) use to plan, design, deliver, and maintain the a software system.

Anything with a lifecycle is waterfall. Not that there is anything wrong with that. Constructing a high rise would be a waterfall project.

But anything that is iterative, like the Amazon app, is not going to benefit from a lifecycle. Its development is ongoing. There is continuous improvement.

According to Merriam-Webster a methodology is 

a body of methods, rules, and postulates employed by a discipline : a particular procedure or set of procedures

That same source defines a life cycle as

a series of stages through which something (such as an individual, culture, or manufactured product) passes during its lifetime

In my opinion, everything has a life cycle but not everything has a methodology.  Software Development is something that can have both.  I would say that the methodology is contained within the life cycle as different phases of the software's life cycle could use different methodologies.  For example software in the sunset stage of its life will probably have different methods applied to it than something that is in the growth stage. So I think you have it backwards.  A methodology will fall within the SDLC. 

Also going to say that agile is not a mindset.  It is a characteristic of an entity. Going back to Merriam-Webster, agile is defined as 

marked by ready ability to move with quick easy grace

//an agile dancer

Sometimes it helps me to understand when I take the dictionary definitions of words and see how they apply to the context in which I am viewing them.  Often words are picked with care to name things.  Then those words become definitions of commercialized activities and the original intent is lost. 

I can recommend reading this article in order to learn more about Agile SDLC.