Did Elinor Ostrom (or others) resolve the tragedy of the commons for common-pool resources? Ostrom (2015: 19) writes:

Individuals or households who have rights to withdraw resource units
from the CPR must be clearly defined, as must the boundaries of the
CPR itself.

Following her definition of goods (see matrix below) I understand that they effectively describe institutional arrangements for organising access to delimitable commons (= high substractability) within delimitable communities (= high exclusion). Consequently they would describe collective action for private goods.
Did they also discuss institutional arrangements for organising access to commons (= high substractability) between communities in a global society (= low exclusion), i.e. for common-pool resources, such as the climate?

Source: Hess &amp; Ostrom (2007)
Depends on your definition of 'resolve'.  In economic there were already proposed solutions for both public good and common-pool resources. However, most of those require some sort of government action, for example Pigouvian taxes/subsidies or in some cases government ownership/provision.
To be more specific before Ostrom and others, literature split goods mainly as either public or private goods (see Samuelson 1954), where it was argued that private goods can be provided by the market and public goods require 'top down' government involvement in one way or the other.
Ostrom's (and her coauthors) contribution was mainly twofold:
First, she showed that actually there is more nuance to this and that there are more dimensions to goods than simply being public or private as the matrix in your question shows (Ostrom 2005). To be fair the category of 'club goods' was already introduced by Buchanan (1965), but she added the idea of subtractability and the fourth category (Ostrom &amp; Ostrom 1977).  Furthermore, she also showed that incentives people face between the public good and common pool resources are different and not similar as previously thought.
Second, she showed that actually once you split public goods into multiple categories, there is a category of common pool resources where 'top-down' government involvement is not necessary and the resources can be managed privately (where by privately in this case is mean to mean without government involvement) or with 'bottom-up' local government (Ostrom 1962; Weschler 1968; Warren 1966; Ostrom 1965, Ostrom 1990, Ostrom 1999, Ostrom 2005). However, her work was not really focused on developing solutions to these problems as much as describing and empirically examining how different communities manage to find solutions to these problems, and those solutions are often case and place specific (for example, the Netherlands where I live has a specific parallel local government for management of dams that is virtually separated from central government with its own local elections and separate structure called Water Boards that would likely qualify as the 'bottom-up' local solution Ostrom was advocating).
However, her work does not really contain specific proposal of resolution, rather as she herself put it in her Nobel Prize lecture:

The most important lesson for public policy analysis derived from the
intellectual journey I have outlined here is that humans have a more
complex motivational structure and more capability to solve social dilemmas
than posited in earlier rational-choice theory. Designing institutions to force
(or nudge) entirely self-interested individuals to achieve better outcomes has
been the major goal posited by policy analysts for governments to accomplish
for much of the past half century. Extensive empirical research leads me to
argue that instead, a core goal of public policy should be to facilitate the development of institutions that bring out the best in humans.

Hence, her solution was to put it simply to just create framework that allows people on local/private level  to find their own unique and specific ways of resolving the problems, which will likely be different depending on local context (e.g. culture etc).
However, I would be hesitant to say that this was a resolution of the tragedy of commons for three reasons:

If you actually take a deep look at the models that study/describe 'resolutions' that arise privately/locally from 'bottom-up' they often show that although this is substantially better than situation without any mechanism or more efficient than government 'top-down' solution they are still not always provided as efficiently as could be done if there would be no such problem (e.g. as in case of private goods). To see that you can have look at the Mueller Public Choice III - the most comprehensive textbook on the public choice (i.e. area of economics that deals with these issues). This is not to say that this would not be the best solution given alternatives, but it does not fully eliminates the problem. To give an analogy if you loose leg, having some modern prosthetic is more efficient than living without leg, but to the extent that the prosthetic is not as good/better as original leg I would be hesitant to call it a resolution of the problem.  In fact Ostrom (1999) herself often describes these solutions as 'coping with the tragedy of commons' rather than 'solving the tragedy of commons'.
Since you wanted to discuss Ostrom, most of the above focuses on her work, but while her work is immensely appreciated, and her Nobel Prize was definitely deserved, it is fair to say that most mainstream economist would argue for much greater top-down government involvement with Pigouvian taxes or cap-and-trade even in cases of common pool resources which she criticized (although here the point 1 would still apply but for different reasons - e.g. it is very hard to know what the optimal Pigouvian tax is).
Rather than developing solution to the tragedy of commons it  was an observation of how people manage to overcome it privately or on local decentralized level of government. This is by no means critique of her work, she was obviously excellent scholar, but saying she resolved the problem would be like saying that Adam Smith resolved the problem of low productivity problem by division of labor, whereas he rather provided an explanation and advocated for something he observed by visiting manufactures and the division of labor was 'invented' as a 'solution' to increase productivity by people not him.