The planner is rubbish and doesn't how to count. I think it automatically adds a day to "go home"… With this itinerary you indeed have 5 travel days.
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EDIT: Make sure to book the Eurostar well in advance. It's a very popular service and there's a passholder quota (pretty much unique). With the Olympics and so on demand will be extremely high.
Look as well at London - Lille-Europe - Lyon so you don't have to transfer stations in Paris. ;)
Lyon - Verona is quite a long journey as the fastest route to Italy is closed until further notice due to a landslide and the main alternative (Simplon) is closed for works from mid-June to mid-September.
The journey is still doable in a day but make sure to leave early. I might consider a Flixbus to Turin to save time.
Looking at a random July day you could do:
- Flixbus Lyon-Perrache - Turin/Torino 08:00 - 12:25
- lunch and visit a bit
- FR Torino Porta Nuova - Verona 15:10 - 17:28 (13€ reservation)
Innsbruck - home is likely a very long journey too. Make sure to leave early and have some buffer before the Eurostar, esp. knowing the German delays…
You could also get the Innsbruck - Amsterdam night train and then go home from there. That would require a 6th pass day (or you buy Verona - Trieste tickets separately).
Feel free to ask anything! :)