Generally speaking, a train ticket from somewhere in the Greater Manchester ticketing area to central Manchester (for example Piccadilly, or Victoria, or Deansgate, or Oxford Road) will have "Manchester CTLZ" as the destination (rather than "Manchester Stns") and thus will allow free use of the Metrolink trams in the Manchester City Centre zone.
Similarly, a train ticket from somewhere on the North side of the Greater Manchester ticketing area (e.g. Littleborough) to somewhere on the South side (e.g. Cheadle Hulme) which necessitates a cross-Manchester City Centre transfer will have routing as "Manchester CTLZ" and so will allow use of the Metrolink trams in the City Centre zone.
However, if one or both of the stations are outside of the Greater Manchester ticketing area, then you have to pay extra for the Metrolink cross-Manchester transfer (Adult single = £1.10).
This is the case; what annoys me is that no single official source (happy to be shown one?) puts it like this. Your cross-Manchester transfer by tram is free if your journey is entirely within the TfGM ticketing area. If you travel by rail into Central Manchester you get free travel in the 'City Zone' too.
It's a good step that the route has been renamed in human-readable terms. It's a baby step at simplification by using common terms in the mess that is TfGM/Metrolink through-ticketing.