As /u/UrAccountabilibuddy says, Americanists increasingly use "enslaver" and "enslaved person", with the usual variants. Master is right out and owning should probably go with it in most contexts. The reasons may sound at first like ick factor, but there's more going on.

The way we speak and write about people in the past inherently informs how we understand them. Since most of our sources are from the enslaving class, the natural human thing to do is repeat most of their usages and hopefully avoid the most egregious ones. The result is that our vocabulary has largely been theirs. We read from the Slaveholder's Lexicon, as it were. That doesn't mean we copy every detail of their thought about things, but the language carries with it the power dynamic. Describing someone as "a slave" who is "owned" by a "master" is factually correct, but a master is not a boss and does not employ people who have a job description interchangeable with that of a secretary, day laborer, or whoever. Referring to them exclusively by the inherited terms carries with it a silent implication that this is both normal and normative.

Both of those implications are true, in important senses, for most of human history but leaving things there ignores the people on the receiving end of them: four million black Americans tortured, whipped, beaten, and raped for profit and pleasure in a civilization-defining institution. By doing that, we are recapitulating the cruelties without comment. To call someone a slave is to, intentionally or otherwise, agree both that that is their legal state and that we should accept it as such. Think of it this way: the Nazis called people subhuman but we don't uncritically repeat that. Why should we uncritically repeat an enslaver's opinion of the people they enslave? We certainly don't use the most common term they employed for black people, which I will not repeat.

Which is closely related to the other big issue with the conventional terms: they're obscurantist. Calling someone a slave ignores that enslaved people can't exist without enslavers. No one is born a slave in some hypothetical state of nature or placed in the situation by the actions if some fundamental force of the universe. People enslave other people and they have very often in the past used the excuse that they were not responsible for the initial act of enslavement, which white Americans feigned and possibly experienced some genuine discomfort over, but that didn't stop them from continuing it. Enslaving people is not, as they argued, a one-time act. The enslaved never want to be, by definition. Rather it's a constant exercise of power, both explicit in the form of horrific violence and implicit in the threat of it that underlies every enslaver-enslaved person interaction. Enslaving isn't a faraway act of capture, but rather among the most intimate kinds of atrocity, reaching into every part of a person's life against their will and against their constant struggle to push back and carve out what tiny slivers of freedom they can. Whether one has a paid overseer or does it oneself, enslaving is a full-time job and every moment of it is another fresh act of life stealing.

It is most certainly our duty as historians to contextualize history. We need to, both in quoting primary sources and in characterizing them, be honest to what they say. That means that the old terms aren't going anywhere, but when used mixed in with those that stress the act of enslaving rather than obscuring it they lose much of their power to harm. As long as one is consistent enough, a good prose stylist can get the job done. By doing this we are contextualizing, specifically in that we introduce into the conventional picture a more fuller understanding. The context, at least for most of us likely to read this post, is one all about white men of high status, to a somewhat lesser degree about all white men, to a significantly lesser still degree about white women, and everyone else is just an object upon white elite white men act.

That context is a fantasy built on horrors. If most white Americans imagine a plantation, they're going to see the beautiful big house, the rows of trees, a nice lawn, gardens, and that kind of thing. It's all quite lovely. Now imagine a slave labor camp, which is what a plantation actually is. Most plantations worth the name are home to significantly more enslaved people than enslavers and they do the actual work of making the thing run. They live too many to a room in ramshackle quarters, or huts, lean-tos, and worse. They grow the food the whites eat. They cook it. They harvest the cotton, tobacco, sugar, wheat, hemp, and all the rest. They tend the livestock. They make and mend the clothes. They raise the white children. They sleep in the same room or just outside it, so they're on hand at a second's notice to serve the whims of their enslavers.

And it's their blood on the end of the whip, their sweat running down the tools, their aching muscles, their tears when their loved ones are stolen away and gone forever, their silent rage and anguish as their parents and their children and their siblings are neglected, starved, beaten, and raped. The enslaver makes all of this so. It is the world that he wanted and worked hard to create, which he loves and celebrates. He deserves tremendous attention for all that, but we must take great care to make sure that we and our readers do not slip into thinking of him as the only person in the story. That's just what he would want. Giving him that, calling him a "master" who "owns" other people plays into the whole dynamic far too neatly. Masters are good at things; one wants mastery of French or American History. We own people when we beat them at games or in debate. They are words of praise and triumph.

Many people always seem to want to celebrate a rich man, endowing him with numerous virtues that explain how he has earned his wealth. He must be smart, innovative, flexible, able to sense opportunity and strike when others miss it. He must be disciplined, thrifty, a person of good taste and comportment. Such a person is, in a word, a gentleman. If we think it of him when we know he's an enslaver, we are calling him gentle, noble, and worthy of emulation while he has his hand on the whip, while the screams rise and blood flies. We name him good as he dismembers families, as he tortures parents in front of their children, as he strips little girls naked and whips them. We make a role model of a monster. All of that is his context. To ignore it, both in the past and how we have often imagined and written of the past, is actually taking him in isolation and depriving us of a full understanding in the service of perfidious mores that oft dare not speak their name.

The new terms are anachronisms, to be sure, but we use them in the service of calling our minds to those parts of history our viciously white supremacist culture has trained us to let slide by unmarked. It is our duty as historians to do better, to call out and correct blindness, willful and otherwise, when we can. We must turn our audiences eyes to those horrors celebrated, ignored, or taken for granted in the lives of eminent men so that we and they might understand our subject and ourselves better .

Increasingly, today's historians who write about chattel slavery or those periods of history use variations on the term "enslaved people" (when referring to groups), "enslaved person/man/woman" or "enslaved child" (when referring to a single person.) Using the term does two things: first, it removes the passive voice that the word "slave" provides and second, it helps the reader think of the person being talked about as an individual. Consider, for example, these two sentences.

• Sarah "Sally" Hemings was a slave owned by President Thomas Jefferson of the United States.
• Sarah "Sally" Hemings, an enslaved woman, was owned by President Thomas Jefferson of the United States.

While they say similar things, the second sentence situates Sally as a person first, property second. When it comes to describing the owner of the enslaved person, practices vary but variations on "owner" or "owned by" as in the example above are increasingly common. I'm not sure what you mean by "forced" to use but I suspect if you started to modify your word choice to better reflective active language, your professors wouldn't object. (Or at least I hope they wouldn't.)

That said, I tend to read about American chattel slavery in the context of education and the word choice I see may be more common among the historians I read. I'll defer to users like /u/freedmenspatrol, though, who do more writing about slavery itself.