Does the term 'protein expression' refer to the production of proteins only or also their regulation?
There are no formal definitions accepted in science for ambiguous
phrases such “protein expression”.

If you wish to know what a particular author who uses such a shorthand phrase means you need to examine the context. If you wish to write scientific reports that are comprehensible to others you must express yourself precisely, defining any shorthand phrases that are necessary to avoid clumsiness.
In the context of the question, all that “protein expression” means to me is “expression of protein” — the synthesis of the protein from some entity that has the potential to do so. It a fundamental level that would be a gene or mRNA that encodes a protein (or proteins — that too is unclear), in which case this might be less ambiguously expressed as, for example:
“…expression of protein from gene X inthe plasmid construct Y…”
“…expression of proteins from mRNA in vitro…”
(I would generally retain the preposition. Be kind to it, poor little thing!)
To indicate that one was interested in the potential of tissues to synthesize proteins, then (with all due respect NCI, who are probably simplifying for the layman) one would need to write something like:
“…expression of protein in liver cells (after prolonged starvation etc.)…”
Under no circumstances would I imagine the term “expression” to imply or include “regulation” or “post-translational modification”. To illustrate why, consider the following example of a perfectly valid use in this sentence:
“A problem with expression of mammalian proteins in bacteria is that the required post-translational modification does not occur.”
If I wished to indicate I was also writing about regulation I would say so specifically:
“The expression of haemoglobin in reticulocytes and its regulation”
(and expect others to do so too.)
The fact that, as a professional molecular biologist, I disagree with @MattDMo, who is likewise a professional scientist, only demonstrates that the term is ambiguous. We would both like the context.