I think Terri didn't want Kyron living there. IMO, she probably never anticipated a lifetime of being Kyron's full time mother, as when she met Kaine and they started their relationship, Desiree had primary custody.
Terri made it clear in her email that she felt she was forced to not work so she could be a stay at home mom for Kyron, blowing through all her savings to do so because "his natural mom wouldn't". She was resentful of Kyron and Desiree and Kaine, that was clear in her own words. IMO, Terri probably thought when Desiree became healthy enough to care for Kyron again, she would. But Kaine wouldn't give up his child, so there she was.
She had no money, and in her own words the reason she had no money was because of Kyron whose presence as a baby forced her to stay home, Desiree who "wouldn't" stay home with Kyron, and Kaine who made her contribute financially to her household. She blamed all 3 of them for her lack of money - those are words written by her that we've been allowed to read.
So with that resentment inside her, I wouldn't be at all surprised if she allowed Kyron to call Desiree and cry, and to tell Desiree that Kyron would be happier living with her, because she simply didn't want to live with this child she resented and she wanted Desiree to take responsibility for him (the responsibility Terri said Desiree "wouldn't" take for him when Kyron was younger).
I also wouldn't be surprised if down the line we discover that in some emails Terri found a way to blame Kaine for some of Kyron's issues, or for tension in the home that led to Kyron wanting to live with Desiree, etc. From all we've read of Terri's own words, she liked to blame others while elevating herself. Would it be out of the character she's shown in previous emails to hear her say that Kaine was the real problem in Kyron's life and that she was suggesting he live with Desiree "for his own good"? I don't think so....
Anyway, I think in some contexts it would be a loving step mother who encouraged a child to call his bio-mother when he was having a tough day and even went so far as to lovingly "give him up" to another household if his distress was extreme... but we haven't seen anything yet from Terri to suggest she was such a step mother. We've read resentment, persecution, a little martyrdom, but certainly not selflessness or overwhelming generous love. She resented him, and in THAT context, telling Desiree she felt Kyron should live with her, followed by emails suggesting "hatred", followed by the knowledge Kaine wouldn't allow Kyron to leave her home, followed by Kyron's mysterious disappearance... yeah.