So I followed the tutorial on making a segue here: https://developer.apple.com/library/ios/recipes/xcode_help-IB_storyboard/chapters/StoryboardSegue.html
When I Ctrl-drag from a button to a view controller I get the options Push, Modal, Popover, Replace and Custom. The tutorial states clearly that all of those are deprecated (except custom, which I guess is for more complicated segues). The deployment target of my project is 8.3 and I use xcode 6.4.
Could someone please shine some brightness unto this confusing mess?
I would strongly recommend you read the documents from Apple about the differences; this document more explicitly explains the differences: Apple Segue Compatibility
These legacy Segues are your Push, Modal, Popover and Replace.
When Xcode 6 was released, the newer functions Show, Show Detail, Present Modally, and Present as Popover were released.
These new functions are designed to replace the old, and are targeted for iOS Versions 8 onwards, but are capable of function in earlier versions, as they are adaptive.
The details of the segues are stated exactly as they are in the Apple Segue Doc that you attached. The document tells you that the newer segues are adaptive. Show has a few more features, presumable for a more flawless operation, quite possibly in regards to conserving memory/preventing memory usage spikes at run time.
In the Apple Segue Doc (the first link), the comparisons are broken down as shown below (there is another image on the site):
If you plan on releasing applications to devices with versions below iOS 8.0, you could consider using the legacy segues, however, if you are only planning on releasing to devices compatible with 8.0 onwards, Show, Show Detail, etc.