Well let's start from the beginning for better visibility - first of all we all have access to that sheet so everyone can pas their own data (phone numbers and text to send) as needed.
One of us who deployed this script based on tutorial linked somewhere above with settings that he (no user who uses app) is one who executes it no matter what. Then when using app we've noticed two cases of what is the result:
Case no.1 user who deployed that script sends messages. These are being sent and delivered correctly. Copies of these messags are being saved on his phone i messages app as he would send them the usual way by just typing it in via messaging app.
Case no.2 anybody else sends messages. Messages are being sent and delivered also correctly, BUT copies of messages sent are not stored in messaging app ie. there is no proof that message was sent other than status in google sheet. While it may be sufficient when one of the recipients asks you back after what was sent to him. For example let's assume that message one sent via this app is as follows:
'The code for you is 13579 Please respond with same code in reverse order for validation purposes'.
Then you get response from that number '97531'. But only his response is in messaging app. Message that was sent by you to that number is missing so you basically are unable to compare response with what was sent.
So we've been digging why is that so that only one of us (one who did deployment and is marked as one who executes script) only have message copies stored on his device while others are sending messages but without copies stored on their phones. I found out that it was connected probably with that in edit history of that sheet status updates were always set by owner's account - even when he was offline. That pushed me to think about it that only thing that is needed to be changed here is to change 'Execute the app' from 'Me' to 'anyone who use webapp'. But then those doctype errors pop up.
I've set another account and when I fetch and send messages with that account it sends messages normally but copies are still saved on my phone [ie. I don't see what was sent when checking phone]. Only change here is that status is now updated by that new account. Not sure at which point where that 'save copy or not' thing is happening.