Does the Enroute Phase begin at a 30NM distance from the departure airport?
You are not getting anything wrong, I think you are simply overthinking something that isn’t actually important.  The 30nm distinction is somewhat arbitrary, and presumably captures most arrival and departure procedures.  There is nothing magical or binding in this number, and there is nothing pilots will do differently when flying a published route or procedure based on distance or phase of flight.   

For example, let’s say the last point on a published departure procedure is less than 30nm from your departure airfield, and the next point on your filed route of flight is outside of 30nm.  Your question implies that there would be confusion as to whether you would be in the terminal phase or the enroute phase of flight as you flew between these two waypoints, is that basically correct?  My point is that this distinction does not matter.  You will not fly the aircraft any differently if it is one or the other, you will simply navigate to the next point on your route while climbing or maintain assigned altitude.  

Now there are some things that are accomplished during arrival and departure phases that may be worth discussing to clarify things and satisfy your curiosity on the subject.  However, most of them are done as a result of passing a certain altitude.  For example, at a certain altitude during climb out power may be adjusted for noise abatement and efficiency.  Passing through 18,000 feet altimeters will be adjusted to 29.92 and perhaps switched into a different mode.  This will be captured as part of a climb checklist.  Reaching cruise altitude another checklist will be accomplish that will include things like adjusting cruise power, checking cabin pressurization, etc.

Preparing for arrival similar descent and approach checklists are executed that will include obtaining weather and local altimeter setting, approach briefing, etc.  Passing 18,000 on the descent altimeters will be set to local setting, passing 10,000 speed will need to be slowed to 250KIAS.  Checklists can be triggered by either altitude or distance from the destination, but again the distinction between exiting the enroute phase and entering the terminal phase is not really that important.  The thing that matters is understanding when you are established on the published procedure you have been cleared for.  The actual distance from the airport where this actually occurs is of no real concern as long as you fly what you are cleared for. 

If IFR there will also be a hand off at some point between the Air Route Traffic Control Center and the local approach controller, but this will likely be tied to altitude as well.  Also, you may be handed from one approach controller to another, but this will occur as you cross sector boundaries and is based more in geographic radar coverage and not a concentric range ring from the airport.

Finally, most modern GPS navigation systems will automatically change the sensitivity of the course deviation indicator in different phases of flight with the least sensitivity in enroute phase, more in terminal, and greater yet in approach phase.  (This is basically what the section you quoted above is referring to)  However, they will cycle as the aircraft passes the first or last waypoint of a procedure that is loaded into the system and not at some distance in between.  In other words, if you are navigating to the first point of a STAR or instrument approach the system ought to retain enroute CDI sensitivity until this point is reached.  The upon passing the waypoint it will cycle to the greater sensitivity of the terminal or approach phase.  For this reason is it important to understand what phase of flight your GPS considers you to be in, but know that it will not cycle at 30nm or any other arbitrary distance in between, but at the waypoint where the transition occurs.   

To reiterate, there isn’t a specific checklist or different way of flying related to the transition between terminal and enroute phase of flight, nor is there any such thing as a “no man’s land” in between phases if a point on a SID or STAR is within or outside of the 30nm distance referenced in your question.