To me, one of the purposes of texting instead of calling is so the recipient doesn't feel obligated to answer immediately.
Though that might be a side effect of me consistently finding guys who might not answer a text for two or three DAYS, because if they receive a text and aren't immediately able to answer they tend to forget they've received a text at all. That's been a problem with each of the three guys I've been with since the polyamory thing started for me. With each of them, I told them that while I'll never make demands of them, if a text goes unanswered for more than a day, because of my anxiety I start wondering if I've done something wrong and they're ghosting on me, so I prefer getting a reply at least the day that I send the text. Boots is making a point to check his phone more frequently since I asked him to at least answer *questions* the same day, and he now answers nearly all of my texts within an hour or two. My two previous boyfriends addressed it in other ways.
But on the other hand... when I was seeing Guy, he was still traveling extensively for work, but was going home ("home" being the place he was renting for his second ex-wife and their son; Guy couldn't afford separate housing for himself so stayed with them when he wasn't on the road) for about 5 days every month. His ex didn't deal well with him talking to other people when he was there, so he would tell me when he was going home and I would make sure I *didn't* text him during those days. I knew he probably wouldn't be able to answer, so it was easier to just leave him alone. The only times I texted when he was home was if I was having a severe anxiety attack or a PTSD issue, because Guy was better able to help me through those than Hubby, and he had told me it was okay to text him under those circumstances as long as I understood that he wouldn't be able to answer right away. On those occasions, he would find something he needed to go to the store for so he could get out of the house to call me.
With Boots, because he's also involved with Glow, I've told both of them that I will not text him when I know they're together. They usually have two separate nights a week together; anything I want to say to him can wait until the following day when he's at work. (He is able to answer texts off and on during his work day.) Boots and Glow have both told me they don't have a problem with me texting him while they're together, because they know I won't abuse it; I don't text him every day as it is, and when I do it's usually just a "hi, how's it going" or a "are we still on for (whatever plan)". But to me, because they aren't together all the time, it just feels disrespectful to both of them if I'm texting him at those times.
So... the short version is yes, I think it's completely reasonable for you and Farmer to set the boundary with Red that, while it's okay for her to text you or Farmer when you're together, she needs to accept that she might not get an immediate response. And I think it is *not* reasonable for her to ask to be told when you're away from your phones. She should be able to understand that sometimes you're not going to answer right away. No one answers their texts immediately every single time they get one, as far as I know...