In terms of computability theory, what is a programming language?
https://en.wikipedia.org/wiki/Turing_completeness says
In computability theory, a system of data-manipulation rules (such as a computer's instruction set, a programming language, or a cellular automaton) is said to be Turing-complete or computationally universal if it can be used to simulate any Turing machine. ... Virtually all programming languages today are Turing-complete.
A universal Turing machine takes in two inputs:
code that encodes a TM
input to that TM
Is a programming language a language used for encoding TMs for providing the first input to a universal TM?
Is an implementation of a programming language, such as an interpreter for the language, a universal TM?
To both of your questions, the answer is "no".
Computability Theory is really the wrong lens for studying Programming Languages. A Turing Machine is a particular programming language, just like Python or Haskell or C. It was just the first programming language to be mathematically formalized in a way that convinced the mathematical community that any "algorithm" could be computed using it. Really this is a point of philosophy. There is no way to prove every algorithm can be computed by a TM, because there is no mathematical definition of "algorithm". Turing argued (extremely convincingly) that our squishy intuitive understanding coincides with the mathematically rigorous notion of TM, and so we bite the bullet and define "algorithm" to be a piece of TM code.
The beauty of Computability Theory is that this (seemingly arbitrary) definition is extremely rigid. Basically any decent model of computation you can think of leads to the exact same notion of algorithm!
A "programming language", then, has nothing to do with Turing Machines. It is just another model of computation. The language is "Turing Complete" whenever it gives the same notion of "algorithm" as Turing Machines! Similarly, an implementation of a programming language has (a priori) nothing to do with universal turing machines. Though you can (and people have) write compilers to turing machines if you want to.
In general, if you want to study programming languages formally, I would recommend Bob Harper's "Practical Foundations of Programming Languages", which was a great resource when I was first learning. It is available for free on his website.
I hope this helps ^_^