Why do prokaryotes not have membrane-bound organelles

Why is it said that bacteria do not have membrane-bound organelles when they often have one or more flagella?

First, this is of some relevance.

https://www.karger.com/Book/Home/258738 says: "The traditional view of biology divides living organisms into two main groups, the eukaryotes and the prokaryotes, the former having membrane-bound organelles but the latter not having recent studies reveal that this view is obviously wrong. "(So this is a link confirming one of the things mentioned in the question.)

Although a knowledgeable biologist I spoke to actually said that the above-mentioned distinction between prokaryotes and eukaryotes is wrong, there are still distinctions, a genetic distinction (nowadays the distinction would be genetic) and also an observable distinction, the prokaryotes don't make got a nucleus and eukaryotes do. So the terms prokaryotes and eukaryotes are still relevant and not in error.

It is possible that texts that read "membrane-bound organelles" mean "membrane-bound organelles in the cytoplasm". However, as I mention below, the flagellum of a bacterium does not appear to have a membrane.

Now to answer the question itself -

I spoke to a biophysicist familiar with microbiology who seems pretty knowledgeable. And he gave an answer that got straight to the point.

He said that the bacterial flagellum has no membrane and none of its intricate internal components have any membranes.

What holds them together (since it is not a membrane) - all flagella material is protein, held together with protein-protein interactions, the same material that holds all protein complexes together, like salt bridges, hydrophobic effect, etc.

The question also mentions this

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3386205/ "Cilia and flagella ... are membrane-bound organelles with a unique membrane."

It seems that the above statement is false or misleading, see

https://www.ncbi.nlm.nih.gov/pubmed/7378551

The term flagella is ambiguous. It refers to bacterial structures made from flagellin protein and to eukaryotic structures made from microtubule proteins and ATPase (tubulin and dynein). The fact that cilia are almost identical to eukaryotic flagella and have nothing in common with prokaryotic flagella From the Terminology is not apparent. It is suggested to adopt Smagina's 30-year-old suggestion repeated by Kuznicki and others: cilia and eukaryotic flagella are called “undulipodia.” The term flagella should be limited to prokaryotic organelles, bacterial flagella - and spirochete axial filaments: solid structures made of flagellin that protrude through the plasma membrane and have no intrinsic motility over their entire length. Undulipodia are defined as intrinsically mobile intracellular structures that show a 9-fold symmetry in the pattern of the arrangement of 24 nm diameter Microt ubuli. They are limited to eukaryotes, members of the protoctist, animal, and the vegetable kingdoms. "

The knowledgeable biologist I spoke to at the time said that nowadays the distinction between eukaryotes and prokaryotes is mainly made by genetics. Nevertheless (as mentioned) an observable distinction between eukaryotes and prokaryotes is that eukaryotes have a nucleus and prokaryotes do not have a nucleus. If a prokaryote was ever discovered to have a nucleus, it would require a reorganization of the phylogenetic trees. While phylogenetic trees use the term eukaryote, they do not use the term prokaryote, although the term prokaryote is still relevant, and there is an organization called the ICSP (International Committee on the Systematics of Prokaryotes) that is on the subject of naming groups of Prokaryotes.

While it's not entirely clear to me if at any point there was a shift from observable to genetic traits to help determine the difference between eukaryotes and prokaryotes, this page mentions something about the origins of the terms prokaryote and eukaryote. And I understand from the knowledgeable biologist I spoke to that these days the distinction is genetic, or mostly genetic. http://www.madsci.org/posts/archives/2000-04/956011333.Gb.r.html

added

I spoke to another microbiologist. He thought the term prokaryote was not useful, it had no evolutionary basis. And at least the term eukaryotes, cells with a nucleus, the cells could be quite related. While prokaryotes - cells without a nucleus - partially define something by what it is not, and it does not phylogenetically / evolutionarily group organisms. Prokaryote refers to bacteria and archaea, both of which are generally unicellular. But archaea are evolutionarily and biochemically more similar to us (eukaryotes) than bacteria. There's no good reason for them to clump together with bacteria. Not evolutionary or biological. The most recent development is that eukaryotes likely evolved from archaea. Some newer models show the Eukarya lineage as rooted in the archaic lineage. All of them show that they are closer to bacteria than either, although this is generally accepted phylogenetically and also cytologically. The three areas of life are bacteria, archaea and eukarya. Eukaryotes still make sense as a grouping because they are a group of organisms that are more similar to themselves than bacteria or archaea. Indeed, they are genetically identified.

James

There are some interesting topics to discuss here, especially in the discussion with the microbiologist about how prokaryotes are often an unhelpful term for describing evolutionary distance. The key point is. This is already included in the short answer from @ GerardoFurtado.

Barlop

@ James I notice Gerardo wrote, "It is clear that prokaryotic flagellum is just a structure made of proteins that crosses the membrane and grows outside of it." So what is a membrane that distinguishes it from just protein is a membrane ... protein + fatty acid? Or is the definition more abstract than something that surrounds something, so in a sense the body has a membrane that is skin?