Hi NFOR and welcome
Your experiments with the balm are very much aligned with the work that has been done using capsaicin pepper to abort CHs. This has been published in multiple medical research journals and articles.
You can read up about some of this research at - Multimedia File Viewing and Clickable Links are available for Registered Members only!! You need to

or

.
The PubMed webist (Multimedia File Viewing and Clickable Links are available for Registered Members only!! You need to

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) is an excellent resource for searching for published research around CH and anything else medical related. It is one of my "go to" resources. Whilst for many articles you are limited to viewing the abstract and not the full article, the abstract should be a good summary of the article. However if you have access to someone who is at a university then they are likely to be able to view the full articles.
Your thoughts around the linkage between CH, TN and ON are around the symptoms they produce. You might find it interesting to look at the info on the IHS's (International Headache Society) website (Multimedia File Viewing and Clickable Links are available for Registered Members only!! You need to

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). The IHS is the international organization which classifies headaches and produces the standard diagnostic criteria for each.
You'll find that there are multiple headaches (Multimedia File Viewing and Clickable Links are available for Registered Members only!! You need to

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) which are linked to the trigeminal nerve.
What you have noted is that the pain symptoms can be very similar between the different headache types. This is what can make diagnosing complex headaches like this to be very difficult as whilst it is fairly straightforward to match the symptoms, the final diagnosis is dependent upon ruling out all other possible causes. This is why we always direct people to a headache specialist since family doctors and even most neurologists simply do not have the skills or experience to do this final step.
Do note that the IHS classify CH as being a primary headache, i.e. it is not caused by anything else, so it is considered to not be caused by TN or ON.
The difference is in what is kicking off the process which results in the symptoms seen. For CH it has been observed that this starts off in the hypothalamus area of the brain.
Please do keep reading and asking questions. It is great to see someone who has been learning as much as they can about CH and other similar headache types.