A Quick Text message is sent from your phone just like a regular text message. To see the Quick Text choices just press the lightning bolt icon when entering text. You can then send or cancel the message.
to see your "reason" you may need to rearrange the messages ("Leg Injury-Can't continue". or "Can't continue leg injury."). I tested on mine using "Going to be late but OK so will contact you later" and it showed two lines "Going to be late but OK so..." which should seem to be enough for you to see the first part of your reason but I don't know how it formats for display on the inReach. Same applies to the 3 preset messages. As SV_TED explained, you can see all the messages in Earthmate.
SV_TED, your phrasing "A Quick Text message is sent from your phone just like a regular text message" could be mis-read by some as the message is sent by the inReach.
On the Mini, you can view the quick text by selecting it, then selecting Write Message instead of send. This allows you to view the full text. You can even modify it before sending. That said, this is a lot of extra button clicks. You really are better off rearranging the text so the messages are unique in the first two lines - so you can select directly from the list.
Twolpert, thanks for this info as I was looking for the same thing. But, when I go to "write message" after selecting a quick text I can then only see the last bit of the quick text. Is there a way to be able to scroll through the whole message? I do see that I can scroll back letter by letter to move backwards (using the <-- arrow symbol) in that quick text while in "write message" mode, but this is quite kludgy and as you backspace you still only see part of the message, the end of the message drops off as you scroll backwards (only showing you a few words of the message). The "autocomplete" section of the screen dominates the area. Thanks for any tips you can provide.