They're calling it "internal syntax" to contrast it with the syntax of previous examples, which are written in a semicolons-and-braces syntax.
See the first paragraph of section 4:
We start by describing the syntax and object model of a core language,
which is intended to be simpler than Plaid source code yet be
capable of representing all of the major semantic elements of
It is typical in programming languages research to talk about a real language in terms of a core language model. The real language is often verbose, and it has many features necessary for getting work done that have nothing to do with the specific point of the paper. Given that most languages are developed as extensions of existing languages, it may have inherited its syntactic and/or semantic complexity from the language it's based on. The core language, in contrast, is designed to be really terse and simple, to have only enough features to illustrate the paper's point. For example, in MacQueen's paper on the ML module system, he models ML with a dependently-typed lambda calculus called SOL. ML has
I/O, exceptions, and all sorts of other features; SOL doesn't, because they aren't necessary to explain how the module system works.