Theory in history can mean a lot of things, but it doesn't mean theory as used in the natural sciences. Whether the natural sciences have a "very strict criteria" for something to be a theory is debatable, but I assume you mean, "a hypothesis that can be tested."

Most of what people mean today by theory in history is "critical theory," which is more a form of philosophy than a form of history, but can be adapted to history. So for example, the work of philosopher Michel Foucault plays a big role in much historical use of "theory." Foucault argued, among other things, that "power" in the modern period became embedded in "rational" mindsets and infrastructure systems, as opposed to just being about physical violence or the divine right of kings or other pre-modern approaches to power. So for Foucault the ultimate expression of modern power is a prison in which inmates fear they may be constantly surveilled, and thus act as if they are surveilled (the "panopticon"). Or it might be in the way in which the state created institutions for tracking and measuring people, giving rewards or punishments to people who failed to meet their new standards ("discipline").

So a historian might say, gosh, that's very interesting, I'm going to study something other than what Foucault was studying — like, say, the Holocaust, and ask myself, "does this go with Foucault's notion of power?" And the answers tend to be: "yes, Foucault's notion of power helps us understand this history, and this history in turn helps us understand Foucault's notion of power as well." Occasionally the answer is: "no, Foucault's notion of power doesn't help us, and here's an alternative way to think about power instead." And this is called "doing theory" when you use history — this kind of back-and-forth with larger philosophical-historical theories.

The goal is to have a conversation with "big ideas" (I just used Foucault as an easy and common example), and ideally show that history can and should be used to either elaborate or contradict these big ideas. This isn't really a testable outcome in the sense that is preferred in the natural sciences (none of this is falsifiable, for example, and it's not clear to me that anyone really believes that these theories are building on each other, or that we're at the "best" theory today).

Now one can use "theory" in a looser sense than what I've done here. I have a theory that Truman did not understand that Hiroshima was a city, for example — that's in line with the conventional use of theory, which is to say, I really mean "hypothesis" if I am being strict about it, because I wouldn't claim I think it is testable to the same degree that the sciences desire. History is empirical (it uses evidence) but it is also highly interpretive, because our evidence is often very messy and incomplete. But if someone says they "do theory" in history, the first sense is usually what they mean ("critical theory," which again is a branch of philosophy).

Historiographic schools of thought are not considered to be theories — they're usually called interpretations, or sometimes arguments, but the word theory is not usually used in such a situation.

Theory in history does not necessarily have a strict definition like that in natural science. The theory of history and theory in history are also two distinct things, although they do overlap at certain places. Kenneth Waltz, who is primarily a professor in International Relations, differentiates theories from laws in that the merit of the theory is from their explanatory power instead of the repeatability of a necessary relationship. Theory in history and of history are both not simply a complex amalgamation of laws into a system which can produce precise results. This is especially true for history since we are often left with incomplete information and are dealing with humans who have agency in their actions.

Theory in history is the use of a theoretical framework to explain events or phenomenon that occur in history. The exact nature of the theory depends on the scale at which someone is writing at: if they are looking at things on a societal level, international level, across decades, across centuries, within the same geographical region or across regions. the They seek to put forward the most compelling case for why something occurred. Perhaps the best example to compare would be the theory of evolution in biology. It is widely accepted as fact due to the extensive amount of compelling evidence for it and its strong explanatory power for how species came to be. However, it remains a theory instead of a law since it is not an ironclad relationship with necessary outcomes that are repeatable. The general pattern and principle may persist but the specifics in outcome can and do differ. The same can be said for the use of theory in history. As an example, the study of revolutions in history is aided by a theory of revolutions with historians such as Charles Tilly utilizing this. The theory of why revolutions occur, what affects is course and its outcome and how they are structured are both built from historical examples and used as a means to compare them. The function of these kinds of theory in history is two-fold, they both seek to explain the occurrence of a phenomenon across time, accounting for their differences as well, and enable comparisons.

Still, even excellent theories in history are limited in the sense that there are often exceptions when examples are brought in from further afield and the complex nature of each historical event almost guarantees that there will be some kind of "unique" event or factor giving it a noticeably different character. Here we reach another problem of scale, the large the scope of our theory, either the vaguer some of its components will be or the more exceptions it will have and vice-versa. For example, if we were to form a theory of why the French Revolution occurred, this enables a delve into specific instances as causes. In a way, history itself is theory as in many instances we are seeking to explain why something happened with no definitive way to prove it as the events have all pasted. While history can agree, most of the time, on what are non-causes (i.e. aliens did not cause WWII) and what is part of the causes of an event (i.e. Resources were part of the consideration in compelling nations to go to war in WWII), there are many competing theories on what is the primary cause or a systematic explanation of why something occurred. Coming back to the example of scale, if we were to expand and ask the question of why European Revolutions occur from say, 1700 to 1990, this will require generalisation within the theory to explain which necessarily opens itself up to more problems of exceptions or the problem of not fitting any of the revolutions as a precise explanation. Assertions of causes, therefore, have to become more general, for instance, there must be unrest in a majority of the population as a necessary condition. Of course, this is just one of many statements that come together to form the theory and it is not so simplistic. It does illustrate how reaching for an extreme scale, for instance, something like an explanation for all conflicts in history can lead to a theory that amounts to what can be described as banal or even truisms.

Schools of thought in history and the historiography of areas or topics in history are, in a way, theories of history, although historians don't think of it in that way. They are fundamentally competing (but not necessarily mutually exclusive or contradictory) explanations for the same event(s), perhaps using different pieces of evidence or reinterpreting the same set of evidence. However, as I said earlier, this is a fundamental part of history and its construction so no one quite considers this to mean theory per se. The utilization of theoretical models to compare history or connect history that has a larger time scale would be considered theory in history. Schools of thought, depending on the context, can also be considered to be theories as they favour one mode of explanation over another. The Annales school favours long-term social-economic explanations for events and the Marxist historical tradition has its materialistic conception of history. However, in another context such as Cold War historiography, schools of thought often indicate waves of authors and their position on the cause of the Cold War instead of shared theoretical framework between them. Hence, it is difficult to say as many of these terms differ depending on the context.

On the other hand, theory of history is more akin to the philosophy of history. This is primarily concerned with historical causation, making historical arguments, the nature of historical knowledge and the nature of history itself. The differences in the philosophy of history are most clear when comparing past conceptions of history between each other. For instance, one of the fundamental conceptions of history is whether or not it has an end, which by extension is related to conceptions of time. In the Greek conception of history and time in general, basically there was no "end" to history and that it would continue for literally an eternity with no necessary pattern, the idea coming from Aristotelian philosophy. On the other hand, with the introduction of Christianity into the Roman Empire, this conception shifted with the idea of history to writers like Augustine and Aquinas viewing history as starting with the fall of man and ending with judgement. History now became finite with a conception of when it began and when it would end. There are further ideas like the idea of history being cyclical which was espoused by Humanist and in Chinese history and the idea of progress in history which came about in the Enlightenment. The current idea of what history is now focused much less on these larger more philosophical conceptions and more so settled on being an exploration of the past.

There has also been a change of what are argued to be the drivers of history. An example of this that you might have come across is "Great man theory" which puts forward the idea that the main actors that can affect historical change are a select few exceptional individuals in positions of great power. The Annales school that I mentioned before also falls into the theory of history since they view long-term social causes as the main drivers of history that are the best explanation of historical events. There are also other ideas of the driving forces of history such as the idea of fortune, which was espoused historically by writers like Machiavelli, which considers the role of chance events in affecting the flow of history. In addition, for religious thinkers, there is also the idea of providence, that God is the force that is guiding history, often manifest through acts of nature or in chance.

This is just a general overview of the idea of what theory in and of history is. For further reading, you should check out John Lewis Gaddis' The Landscape of History which is a short and informative overview of the theory of history. On the other hand, if you want to look more at the application of theory in history and are interested in revolutions, you can check out the works of Charles Tilly who blends social sciences and history together.