SOLVED
You can use the Forms API -- a background Marketo form submit -- to submit values from a custom form.
Are there any links for that? The only lead I have found was on the Developers Marketo page, but they only show examples with Marketo forms.
Well, it's kind of the same story, since even a minor attack on a REST-based form will bring it down and the rest of your REST-based integrations, too, unless there's actual technical protection.
Okay, I will talk with our web devs regarding this issue. If the Forms API really do work with external custom forms, that would help us out a lot.
So back to the topic, I guess it is not possible that the "Data value changed" trigger works on fields with previous value empty?
It works on fields that change from blank to empty.
It doesn't work on fields that are created non-empty alongside a new lead.
Thanks for clarifying, this basically answers the topic's question 
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