It's dangerous to extrapolate rules from examples. Unfortunately, many examples in the SR5 sourcebook have inconsistencies or plain errors.
Yes, hence my "if correct".
In this case we have a choice between following the text as written, which gives rules for using First Aid untrained, but that heals 0 damage due to the First Aid skill limit, or the example in which that part of the rule isn't mentioned at all.
So really in the absence of errata stating otherwise, one of the following is probably the case for First Aid :
1. Medkit wireless bonus that increases limit also increases that limit, but it wasn't mentioned under medkit.
2. Medkit skill replaces First AId skill on that limit, if higher, but not mentioned.
3. Using First Aid untrained can only result in stabilisation, not healing (but just letting the medkit work unattended can result in healing)
4. Rule on boxes healed dates back to a time when limits weren't in SR5 and should be removed.
Rules as written to me hint at 1 or 3, but aren't clear and are contradictory. 4 is probably simplest.
I suspect the Medkit entry on p450 should just say First Aid AND Medicine.