I was watching a stream at one time (wish I had a link, but it was a Microsoft project), and they called that concept 'source open' (viewable but not not modifiable).

I think the preferred term for software that doesn't comply with the OSI's definition is "source available".

If they own it, in what way do they own it?

"Open Source" is not a strange phrase. It's the natural phrase for software that doesn't keep its source secret (or "closed", a common synonym for secret). This is so natural that the intelligence community has a parallel definition of "open source" that refers to non-secret/publicly available intelligence sources.

I understand that in practice, ESR successfully muscled others out and claimed effective cultural ownership of the phrase, but that doesn't mean that no one else is allowed to try to stake their claim and see if it sticks.

I personally think the unlimited redistribution clauses that are mandatory for "Open Source" licenses have caused some serious problems. Commercial vendors have improperly conflated the distribution of source code with the complete inability to make any money from it.

The problem, of course, is not that the source is available and people are free to tweak and understand what's happening on their own machines, but that the licenses which open-source programs grew up under made serious commercial sales effectively impossible. The only business model that's viable is selling support. Compare the market caps of Oracle and Microsoft to Red Hat for an illustration of the comparative effectiveness of these models.

Stallman at least swapped commercial viability out for the infectious nature of copyleft, intended to keep as much software free and shareable as possible. Permissive BSD/MIT family licenses just give it up.