Interesting. The language of the law only allows appeals on final decisions (which includes injunctive relief but not preliminarily injunctive relief that you incorrectly lumped together)
https://www.law.cornell.edu/uscode/text/28/1291
Case law developed to allow some departure from the strict language, but per SCOTUS the criteria are
As such, this TRO would normally not be appealable.
https://en.wikipedia.org/wiki/Interlocutory_appeal
Let's see if he'll even hear it before camp
I hate to come across elitist, but just like I am not a doctor and should not get into a debate with one about what I found on WebMD, your comparisons and attempted parsing of the law is not correct.
There are three forms of injunctive relief - a TRO which can last 14 days unless extended, a preliminary injunction which issues during a case and can effectively end a controversy and a permanent injunction which will continue beyond the case itself.
All three are forms of injunctive relief with different standards for issuance of an injunction and appealability.
While a final decision or a decision on an injunction is appealable as of right, all three forms of injunctive relief are appealable, but it varies as to whether an appellant will be able to meet the applicable standard. Again, I am not saying this because I found it on google, its because I do this for a living.