Zurich to Vienna - Cheap 1st Class Ticket - DID I DO SOMETHING WRONG

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I need to do a sanity check and make sure I'm not doing anything wrong.

Yesterday I purchased point to point 1st class tickets from Zurich to Vienna for 2 adults and 2 children under 16 on the RJ 169 Train.  The fair I paid was 194 swiss francs through sbb.ch. This seems very inexpensive to me and I hope all I need to show is the tickets that I have printed and my USA passports.  I have no other special discount passes or cards.  I noticed on the receipt that the two adults were paid and that the two children were free.  The receipt showed 2 adult supersaver tickets and my 2 children were called "children with TEE family discount"  Then I noticed that under the conditions of the children is says "must be accompanied by a parent or a grandparent also travelling on a "TEE ticket"".  What is a TEE ticket?  From what I can tell TEE seems to be a program between Germany / Austria and Switzerland and not necessarily some sort of special ticket.

"TEE" is the name of the cross-border price system run jointly by Deutsche Bahn DB, Austrian Railways ÖBB and Swiss Federal Railways SBB.
And yes, between this countries children <16 accompanied by parents travel for free, so basically in your case you have kind of a "family ticket".
The fare 194 CHF is also correct, so don't worry. It's definitely a good price, especially taking into account 1st class and distance between Zurich and Vienna. But the fare is like this because it is so called Supersaver Ticket Europe (Zurich-Vienna 92 CHF per person on the contrary to 149 CHF full fare ticket on this route), thus discounted international ticket sold as special "offer", sold only online and in advance. And since it is an offer, those tickets are not exchangeable/refundable and non-transferable to other person at any time.

Thank you for the clarification.  I also wrote to SBB and they also confirmed that the deal is correct.  Great Forum!  First time traveling in Europe via train.

First time traveling in Europe via train.

Good for you
New railjet train and route via Alps should be a great experience for a first timers.