1) OK, we are getting closer. Since you have 2 separate lights, then most likely the ceiling black wire is connecting to the 2nd light fixture. Splice the ceiling red wire to the ceiling black house wire and to the LED black wire. This wire splice should take care of the 2 switched loop hots that extend to the 2 separate light fixtures.
2) You need a 2 wire AC voltage tester (a contact type or an AC voltmeter) The whites in the ceiling box need to be confirmed if one or more of them are true neutrals. Otherwise it becomes a guessing game as to how the ceiling box was configured since you have Romex cables. White in Romex cables are allowed to be used as either a neutral or as a hot. If white is used as a hot, then they must be re-identified with black electrical tape. Thus, 1 or more of the ceiling whites may either be a hot or a neutral. Your guess is as good as mine. Only a voltage tester can confirm what you have in place for the whites.
3) Since 3 separate Romex cables, I suspect that all 3 ceiling whites are neutrals, but it must be confirmed.
4) If the Klein is a non-contact voltage tester without 2 wires (the kind that only lights up), those are pretty much useless testers when doing electrical work. Whenever working on an electrical project, a 2 wire (contact type) of voltage tester is always required to confirm the presence of voltage. The non-contact testers sense voltage based on induction and these testers can easily provide false readings indicating a hot from another nearby wire. Electricians always have a 2 wire lead AC voltage tester readily available when doing electrical work.
5) The red ceiling house wire is the switched loop wire that extends from the wall switch and will be spliced to the LED black wire and most likely to the other ceiling black house wire (the black is most likely feeding to the 2nd light fixture).