“Rather, it’s a job ad that requests unsolicited applications from job seekers if they currently cannot find a matching job opening at a company.”

Yes, basically you’re correct, though for all intents and purposes, it is an Initiativbewerbung. In British English (could be useful):
https://www.theguardian.com/careers/speculative-applications
https://dictionary.cambridge.org/dictionary/english/speculat...

For example, see here (at the bottom): https://karriere-bewerbung.gleisslutz.com/stellenangebote.ht...

This one is not from a job posting site, but it’s a normal company page. It reads, “Kein passendes Stellenangebot gefunden? Dann erwarten wir gerne Ihre Initiativbewerbung.”

Click the button, and you‘ll get to another page called Initiativstelle: https://karriere-bewerbung.gleisslutz.com/stellenangebot-for...

As you can see on that page and many others, e.g., https://karriere.volkswagen-otlg.de/jobboerse/Initiativstell... , it’s not a generic application. A generic application would be a standardized form, as for a government job (or even for your landlord or landlady!). Here’s an example from WorkSource Washington (which provides services like those offered by the German Arbeitsamt): https://www.aabergsequipment.com/galleryupload/webdoc/Forms/...

This has little to do with the above links, where you can upload both your resume and your cover letter. Like this one:
https://www.whitespot.ca/general-application-form

Besides, there is this:
“‘We had to bin at least seventy percent of all applications because they were either too generic, full of bad grammar and spelling, or they simply did not answer the person specification that we had clearly set our candidates to respond to’.”
http://www.doesntgrowontrees.co.uk/applying-for-your-next-jo...

Or, as they say: http://integritysearchandconsulting.com/2016/02/generic-resu...

An Initiativbewerbung does not mean you send them these kinds of generic documents, unless you want yours to end up in the trash. There seem to be exceptions like Michael’s Brown & Caldwell link; though if you take a closer look, you’ll see that they call it something else(!): http://www.brownandcaldwell.com/CareersFaqs/#q4

Thus, what does work is general application (see above) because you are expressing your “general interest” in a job:
“Don't see a fit in our current open roles, but want to be a part of the amazing OJO team? Send us your resume and a little information about yourself. Make sure to let us know what kind of roles you'd be interested in - and we'll keep you in mind for future openings!”
https://www.indeed.com/q-General-Application-jobs.html?vjk=6...

There are tons and tons of ENS links for this. Another three can be found in the discussion box. Or one from NYC (“General Application at…”):

“We're Growing!
Don't see an opening that's a fit with your experience? Submit your resume here and we'll contact you when a role opens that might be a match.
We've grown by 150% over the past year and are always adding new roles.
If you have experience in Artist Relations, Business Development, Creative, Engineering, Finance, Marketing, Merchandising, Office Services, People Ops, Product Management, or Sales, we want to hear from you.”
https://boards.greenhouse.io/splice/jobs/4101498002

I don’t like “unspecified” that much either. Even though your example may not be really specific (but rather “general”), it isn’t entirely non-specific. It gets even worse here:
https://recruiting.fraunhofer.de/Vacancies/42952/Description...

They’re looking for three types of applicants. One of them: “Sie suchen ein [sic] Praktikumsplatz für ein Pflichtpraktikum.”

That’s actually pretty specific. However, it’s an opening that is not listed or unlisted. See: https://blog.execu-search.com/the-hidden-job-market-how-to-l...
(I prefer the first option because the second one could mean the job just hasn’t been advertised yet.)

An example combining general in the title and not listed in the description:
“Please submit your application here for interest in positions that were not listed.”
https://www.peppermillreno.com/employment/opening/216

You can even find this on job posting sites as a feature:
“There are two ways that candidates can be added to the General Application
• Candidates can apply to your General Application by clicking the button at the top of your Career page. To configure the text of the button, edit your Career page.
…”
https://help.smartrecruiters.com/Interviewing_and_Hiring/Man...

Here's one for the UK:
“General Applications
Can't see an open vacancy that matches your skills? Feel free to send us your CV and we can reach out when relevant roles become available.”
https://olddownestate.co.uk/careers/

But there don't seem to be many British ones. I think “role” may be better in this case:
“Looking for another role not listed?
- Full Time (ALL BRANCH LOCATIONS - Blandford, Bristol, Bournemouth, Bournemouth Airport, Cardiff, Dorchester, Exeter, Fareham, Poole, Salisbury, Southampton, Taunton, Yeovil, Wareham )”
https://www.glassdoor.co.uk/job-listing/looking-for-another-...

Additionally, I don’t quite see how “open application” answers your query. There may be a few examples, but there are also these, which have nothing to do at all with “Initiativ…”:
“If, having looked at the different Colleges, you don't mind which you attend then you don't have to choose — you could make an open application instead.
After the closing date (15 October), a computer program allocates each open applicant to a College.”
https://www.undergraduate.study.cam.ac.uk/applying/ucas-appl...

“Open Applications for the Lab Company 2017-18
Theatre Royal are now open for applications for The Lab Company 2017-2018.”
https://www.barbicantheatre.co.uk/open-applications-lab-comp...

Many of the links are Dutch or Danish or whatever else in origin. I must admit I can’t even figure out why.

In fact, the Initiativ bit in German is not part of the compound noun in English. From my second link in the d-box:

“If you wish to fill out an application and have it be kept on file…”

That is “taking the initiative.”

Considering the Fraunhofer link above, it may sometimes be useful to omit it altogether and create an Internship page or the like (but add that people can also apply for opportunities that have not been listed):
https://mercury.postlight.com/amp?url=https://www.thersa.org...
http://wtonvolunteers.org/choosing-opportunities/apply-here

Here's another idea (see also above):
“Submit your C.V. for future roles
Dearman is always keen to hear from those who would like to join our team, even if it is for a role that is not listed above.”
https://dearman.co.uk/careers/current-vacancies