"HPLC" is not an officially-defined term, so what "defines" an HPLC column is that someone chooses to call it an HPLC column. A bit of Humpty-Dumpty logic there *.
The official term is "Liquid Chromatography" ("LC"). Variants are used to differentiate specific subsets, often more on the basis of marketing than any fundamental difference. Examples include:
HPLC = "High Performance Liquid Chromatography" or "High Pressure Liquid Chromatography"
FPLC = "Fast Protein Liquid Chromatography"
IC = "Ion Chromatography"
UPLC = "Ultra Performance Liquid Chromatography" (that one is trademarked)
UHPLC = "Ultra-High Perormance Liquid Chromatography"
SDPLC = "Super-Duper Performance Liquid Chromatography" (okay, I made that one up just to see if you were paying attention

)
The "HPLC" acronym entered the lexicon in the late 60s / early 70s to distinguish the then-new technology of using columns packed with very small particles to minimize peak broadening during the separation. The catch was that such columns have a high flow resistance and so cannot be used with simple gravity or vacuum-assisted flow.
Today, the term is generally used to describe any "instrumental" version of liquid chromatography utilizing a pressurized system to maintain constant flow and usually a flow-through detector. The monoliths have a lower flow resistance than equivalent packed columns, but the flow resistance is still far too high to operate via gravity and/or vacuum, so most people would include them under the "HPLC" category.
* "
When I use a word," Humpty Dumpty said in rather a scornful tone, "it means just what I choose it to mean -- neither more nor less."
"The question is," said Alice, "whether you can make words mean so many different things."
"The question is," said Humpty Dumpty, "which is to be master - - that's all."
Through the Looking Glass; Lewis Carroll