I will recommend a few books, but first I do want to point out a possible misconception in how you think of history. Very, very few modern historians would describe history as a science. It doesn't really have the markers of science. No ability to conduct experiments or really falsify claims (with some exceptions), little to no ability to construct rational laws based on data, and most importantly -- the kinds of knowledge that we have access to as historians are utterly different from the kinds of knowledge scientists have access to. This is connected to the thing about not being able to experiment, but it runs deeper. As historians, we can observe the past through the writings of others and sometimes through physical artifacts. We must be forever humble and cautious about the fact that we are never going to get the "full story", and in fact we will have immense difficulty even figuring out what a "full story" would look like. In my "day job" as a research scientist, I can observe and make sense of the physical world through more direct means. As scientists we are still humble and cautious about our observations, but it is usually much easier to figure out whether there is a missing piece in what we see. And many of us are at least decently confident that there is a "full picture" to eventually grasp.

Some books that I have found helpful in defining the questions of "what is history, how do we do it", and examining how historians go searching for it. All of these books have been influential in how I understand what history is, and why history happens.

• Marc Bloch, The Historian's Craft. An easy read overall, about 200 pages and written in pretty non-academic prose. Bloch was a member of a French school of history called the Annales, which took a very long view of history. However, this book is fairly generalist and does not especially take sides in debates on "what is history"? It's eminently quotable, too. " The good historian is like the giant of the fairy tale. He knows that wherever he catches the scent of human flesh, there his quarry lies!"
• Ernst Breisach, Historiography: Ancient, Medieval and Modern. This is a really dense, kinda dry read about the history of historiography as it were. It was a textbook for my undergrad historiography course. 3rd ed. is from 2007 so there might be something newer/better as a survey now.
• Alex Statiev, The Soviet Counterinsurgency in the Western Borderlands. Very narrowly focused, but it presents some very interesting frameworks for questions of identities in history. For the "what" and the "why" of history, we find that questions of identity are often overwhelmingly important. Especially in modern history, this collective question of "who am I?" drives conflict, cultural change, and momentous events. And, at a philosophical level, if history is concerned with the study of people then it is quite important to wonder who those people believed they were. I consider this, along with Adam Tooze's Wages of Destruction and Morton and Dahms' The Trial of Tempel Anneke, to be the 3 books that were most influential on me as a budding amateur historian in undergrad.
• Adam Tooze, The Wages of Destruction. This economic history of the German war economy before and during WWII is a magnificent piece of materialist, economic history. When we think about the "why" of history, one of the common answers is to go looking at people's material conditions, and how their societies' economies worked. Tooze makes a great case for this approach, treating the Nazi economy not only as an economy in of itself but as both a source and reflection of Nazi culture-- and he shows his work very well.
• Peter Morton & Barbara Dähms, The Trial of Tempel Anneke: Records of a Witchcraft Trial in Brunswick, Germany, 1663. This is a piece of so-called "microhistory", an approach popularized by Carlo Ginzburg's 1976 The Cheese and the Worms. Microhistory uses records of an individual, relatively small-scale event to tease out information about a society's culture and day to day life. Witchcraft trials are also a great master-class for the budding historian. They are often richly documented by legal paperwork, but the paperwork itself reveals many things by what it does and does not include or emphasize. As Bloch reminds us, "Ordinarily, we prick up our ears far more eagerly when we are permitted to overhear what was never intended to be said," and witch trial records are a rich vein of such revelations.
• EH Carr, What is History? Carr represents a particularly relativist, subjective view of history. The central point that Carr makes is that historians, whether they realize it or not, are themselves "in" the process of history. They decide, in the course of writing and discussion, which pieces of information are merely "facts about the past", and which pieces are elevated as "facts of history". His stance leads more towards your family's position, which you are opposed to, but it is an influential and cogent book and it would do you well to read it (or at least the essay "The Historian and His Facts".) There's a rebuttal to it by Geoffrey Elton called The Practice of History that is probably worth reading too, but it did not make much of an impression on me.
• Fernand Braudel, On History. Braudel was another member of the French Annales school, like Bloch, but he lived long enough to write a much longer book on the topic. Unlike Bloch this is dense and academic in a way that only the French can be. It is a very very good bit of theory work, though.
• Gayatri Spivak, "Can the Subaltern Speak?" This essay is the classic foundation of post-colonial or anti-colonial theory. It's not exactly history on its own, but a lot of people (Spivak included) would make the case that you can't really separate things like colonialism from the study of history. It runs very, very contrary to a lot of Euro-American historiography and should absolutely be read, because even now some 35 years after it was written, postcolonial/decolonial theory is often treated as its own separate bubble.

Are there any books that explain how historians conduct their studies, write their books, and generally come to their conclusions on the subject of history?

Yes, that would be any account of the historical method. Unfortunately, I am not aware of any English-language book which attempts to give a systematic account of the endeavor, so you're probably stuck with whatever others recommend. (In German, I'd direct towards Jörn Rüsen's Historik for one such attempt that I found quite compelling.)

Since you mentioned issues of bias and objectivity in your paragraph about "contrarian arguments," and dropped the term "truthfulness" in there, to be frank: as with any other scholarly endeavor, you decide what is account of it "truthful" and what isn't by rational inquiry.

What historians do can be rather simply described as follows:

Historians present a proposal of how to think about the past in the form of a rational argument, which is the principal result of their historical research. [not a quotation btw, I'm just using this formatting for emphasis]

This means that we need to give reasons which are rationally defensible for everything we say. This also means that we are actually discussing not issues of truth first-and-foremost, but of plausibility. As u/Jon_Beveryman suggested already, the usefulness of "scientific objectivity" in history more or less ends with statements like "a document exists which reads as follows and some chemical tests suggest that it is from the 13th century," which is rather trivial. (It should be noted that philosophers tried to make history, and with it all the humanities, into a "proper" science for at good part of the 19th and 20th centuries until they eventually just gave up in light of the sheer impossibility.)

But anyway,

Since history is typically presented as a text ("narrative"), we can analytically cut up this text into constituent parts in order to assess it. Questions that come into play here are whether the sources referenced are represented accurately (truthfully!), whether they are appropriate and adequate for the argument, and whether some that should have been used have been omitted. We can also analyze the rhetorical dimension of the text and ask whether the causal relationships between single events, the relevance of documents, the value judgments given, are plausible, and we can check for logical contradictions.

Lastly, since it is very rare that only one account about a subject matter exists nowadays, we also can (and should) compare the accounts with other accounts about the same, or a closely related, topic, and see how they fare against each other. Of course, since we are talking about narratives, we can apply the same criteria to whatever some religious doctrine claims about the past, or what a politician in their speech uses to justify their acts, or what to think of outlandish things like straight-up denials of the Roman Empire's existence on TikTok.

Of course, a historian's output also has a certain aesthetic quality built into it (language is intrinsically metaphorical) which is irreducible to the mere "factuality" of things. But this is inevitable, and arguably intended, since we try to make sense of the past, and not produce a nomothetical description of it. It follows that you cannot reduce history to some mathematical equation which ends up being either true or false, but you can use the cognitive facilities given to us by virtue of being human beings capable of rational thought to judge the plausibility, and with it the relative quality, of any narrative about the past.

Discussing these problems more in-depth is the domain of philosophy of history, specifically of epistemology (the inquiry into what knowledge is and how we can know). If you happen to have a baseline education and/or a high tolerance for jargon, a recent scholarly book on this side of things which I found useful is Jouni-Matti Kuukkanen's Postnarrativist Philosophy of Historiography (2015). As all purely scholarly publications, its expensive, though.

And to add another book recommendation to the batch already given, Chiel van den Akker's The Modern Idea of History: An Introduction (2020) is a concise overview of the whole endeavor of history over the past two-hundred years (and some of the problems that we had and have to face), intended as a textbook. I thought it was pretty good.

Making History, Lambert and Schofield

It sounds like you’re looking for the field of Historiography, which is more the history of history, rather than the science of history. Look for publications in the field that stand out to you, but Gaddis is a great start. It has been awhile since I’ve read TLOH, but I am sure the sources used by Gaddis will also be a useful jumping off point for your interests.

The other comments already answer your general question sufficiently, but I can add another book recommendation to the list. (And like others mention, what you seem to be looking for are books related to historiography rather than to the "science of history." Historiography is essentially the study of the development of historical writing, analysis of changing meta-narratives in a given historical field over time, etc.)

Doing History: An Introduction to the Historian's Craft (2015, Oxford University Press) by Pojman et al. was a book I used in undergrad for a historical methods course that would likely be of interest to you. It provides an overview of what historians actually do and highlights some of the various approaches and methodologies regularly employed in the discipline.