I have 2 theories as to what happened...neither are happy but they are valid in that they have some facts to base them on
1) Elizabeth killed Gabriel in a fit of rage....a valid premise given her past rage, her mental problems, off her "meds"...anger management classes,
she didn't seem to love Gabriel, called him "thing", had neglected him....she may have been frustrated when her attempt to adopt him out thru TS failed
she seemed very angry at Logan and may have taken that rage out on Gabriel
2) Elizabeth may have overdosed Gabriel...not really meaning to kill him but meaning to knock him out for her own selfish reasons. I think this is a very likely theory. The pictures of Gabriel in the motel in San Antonio...the "tyelenol"...the words of the "babysitter"
also a baby who overdoses would likely be "blue"....fits with her text. Elizabeth described Gabe as "blue">> she did not say "bloody" or "battered" or "bruised" or "messed up" etc...she said "blue": to me that would be consistent with an overdose
The lack of physical evidence in the motel room ....ie, blood, signs of violence ...also lead me to this theory
I think it would more likely have been rage. Elizabeth has a long history - from childhood - of uncontrolled rage including violence (chased dad around with an object hitting him as a child). Reports indicate that she is easily frustrated, and easily gets to the boiling point and then strikes out.
She seems to me to single-mindedly, unreasonably, impatiently, and selfishly pursue her wants, easily frustrated when she doesn't get instant gratification.
That week in San Antonio, on the heels of the week in Tempe, may have presented a slow build up to not getting what she wanted, with an accompanied waffling between thinking she was going to and being frustrated that she wasn't.
The 26th was accompanied by multiple phone calls from multiple people telling she couldn't, shouldn't, do what she wanted to do - from Tammi, Tammi's attorney friend, and Tammi's friend JM. I can easily see how the evening/night of the 26th could be the final boiling point.
Likewise, she may have gone to bed the night of the 26th thinking everything was going to work out, but have the frustration and irritation of all those phone calls, all those people telling her what to do under skin. Perhaps on the morning of the 27th she did plan on the exchange of Gabe with the couple, and either went to the park to meet them or didn't, but either way, at the last minute, around 10am, found out her plans were not going to work.
It is a fact that shortly after the time the exchange was to take place, that rather than being happy and ready to go on her trip, she was on the phone texting and talking to Logan, in a highly distraught state, making extremely vicious claims of having just killed Gabe. It is difficult to reconcile that as an act to hurt Logan if she really did just successfully accomplish effecting a solution to her problems.
An aside - regards physical evidence in the hotel - I personally think that if Elizabeth did kill Gabe, that she most likely did not do it in the hotel room. I don't want to elaborate on my thoughts on that just now, but I will say that one of my reasons for thinking that is her mention of a park. A park was on her mind, and that leads me to think that it is likely that she killed Gabe in a park or park-like setting - forest, woods, etc.