How to group view by field and display row results below?
What you're trying to achieve isn't grouping by a field, it is displaying the same node list twice, showing different node information in the two instances. 

You can achieve it by creating two Displays in your Views where one is showing only Page Type field, and the other is showing Title and field_description while sharing the same Filters. 

If your original View is a Page Display, it makes sense to use Attachment Display for the other. You adjust the Attachment to appear Before or After the other Display results, depending on which you created first. You could also make one or both of them Block Displays and place them one below the other in the Content region of your theme. 

You could also render one View Display into another Display Header, with the help of an additional module Views Field View. 

Your HTML won't be exactly the same as in your question, but it will be close enough, and you can override the templates in your theme for those particular views displays to achieve exactly the output you want. Attachment views usually show with their own class and view-rows outside the .views-content of the Display they're attached to, but if you really need to you can override them and set your own classes. 

Drawing from Drupal 7 solution

I posted an answer to a similar question for Drupal 7: Filter a view by clicking on a link on another view. It doesn't completely mirror to D8 but with Better Exposed filters you can make an exposed filter behave similar to tabs, if you use Autosubmit optin and hide the Submit button. 

Taxonomy summary through a Contextual filter

One additional way of listing only the terms used on nodes in Views results is by creating a Summary through a Contextual filter. 


On your Content Views Page add a Relationship to the Page Type field, you may require it since none of your results will have it empty. 
Then add a Contextual filter Term ID from Taxonomy term Category and set it to use the Relationship from above, and these options:
When the filter value is NOT in the URL
Display a summary
Descending
Number of records
Base path (if this is not the first Page Display in your Views)
Add Attachment View Display to this Page, keep all the options the same as in Page, only override the Contextual filter and remove it from Attachment. 


This way you get the original list of nodes, after the terms relevant to the results. 

The terms are already sorted by the number of results tagged with it. To make the nodes in the Attachment sort the same way use Aggregation on it.