Trying to understand Search and Factbook

Search and Factbook all search "datasets" that Logos builds and do not actually search the books in my library. Is that correct????

You can get an idea of how Factbook works from the Logos Help resource. First thing is that it does not require use of the Internet, so it is generated on your computer from your resources and datasets. This give you an idea of the datasets it has to access:-

Search can also use these datasets, and there are many other other 'topical' datasets. A <Person ....> search will use Biblical-Referents + Biblical People and 'map' the results onto your bible provided they have a reverse interlinear. In general, a term that uses {...} will use datasets, but terms like <Bible Jn 3>, <TDNT 8.1> are datatypes found in a resource via tagging.  The first is a bible reference, the second is a reference to an entry in the Theological Dictionary of the New Testament. Strong's Numbers are another datatype e.g. <G875>, <H765>  (abbreviated). See https://wiki.logos.com/Search_HELP#id:1.2e_References

I can understand the advantages of searching the datasets to be faster, etc

Speed is not the reason. It is the ability to store complex data entities so that they can be mapped onto many different bibles (for example) without having to tag every bible for the same data e.g. Cultural Concepts, Biblical Events.

And is the building of the datasets a manual process or a program generated one?  

That depends on the dataset. The process for building it is included in its documentation. 

So Logos people build these datasets... and (for what it's worth)... then Factbook and Search hit the datasets to pull data they have been requested to find??

Faithlife build the datasets and they are classed as resources because you have to purchase & download them, usually as part of a Feature Upgrade as opposed to a Library/Collection of books. Datasets do not appear in your Library.

Then Factbook and Search can access the datasets to get to the data that has been requested!

Uh Huh!  So if someone doesn't buy the feature upgrade... then their ability to search is limited because they don't have access to the right "datasets"???  Am I thinking right??

That is correct.

For example, one of the datasets Dave showed above is for Biblical People. This enables us to search for something like <Person Paul> and find every occurrence where that person is mentioned even if not by name.

If you don’t have that dataset, you can’t do that search.

How do I tell which and how many datasets I have?

Click the icon with three dots at the top-right of the Logos screen and select the About Logos Bible software option from the menu

Then are are a list of datasets in the scrolling panel on the left

Edit: And how do I tell what I am missing?

Try using this url - https://www.logos.com/search?sortBy=Relevance&limit=60&page=1&ownership=unowned&geographicAvailability=availableToMe&filters=resourcetype-datasets_Resource%20Type 

It filters the Logos sales page to just show datasets that are unowned by you.

Alternatively, go to https://www.logos.com/compare/featuresets and tick the Hide features I own button to see which features you don't have access to. This might be more useful than looking at datasets as it tries to show specifically what functionality you don't currently have available without needing to know which datasets support which particular functionality.