As we all know, fraud nullifies an election. As such, legally, NO election occured in last November. The US government failed to hold a legitimate election to the highest office in the country.
As such, when President Trump's term ended at noon on the 20th of January, there was no elected successor.
What happens when an office is suddenly vacant because the term of its holder runs out but no other person is inaugurated to it? Especially when said office and the powers associated with it are crucial to the function of the state? The United States could, as far as I know, NOT function without a person who is authorized to yield the powers of President. The United States cannot wage war when nobody is authorized to use nuke codes. The United States cannot react to emergencies when nobody is authorized to declare one on federal level.
As such, somebody has to inherit, if not the office of President, at least the powers and privileges associated with it, becoming Acting President of the United States.
As the events of last November are, legally, not an election, and the event of January 20th was not, legally, an Inauguration, NOBODY currently actually occupies the office of President. However, there must be an Acting President, and acting officeholders.
NOBODY won the election because there has not been an election. A fradulent election is not an election at all. And yet, the terms of President Trump and all officeholders appointed by him ran out.
I believe that in the situation of the absence of a legitimate new heir, any scholar would agree that the person holding an office loses the office but inherits its powers from himself when the term ends.
As such, President Trump ceased to be President of the United States at noon on January 20th and immediately became Acting President of the United States, as by virtue of having been the last legitimately elected President, he is the person most qualified to inherit the powers of the Presidency.
He is basically what in monarchies is a Regent, the only differences from a normal President being the fact that he is obliged to organize a contingent election as soon as possible and that the date of this election and of the inauguration of the next legitimate President (i.e. return of President Trump to the full, not "Acting" office) need not be in October and January.
Other than that, I think he is at least equal to a President who was VP and is serving the remaining term of a President who resigned or died.
He should be able to give executive orders, sign laws, command the military, appoint and fire people (just add "Acting" to the office designation). He might have to countersign his own orders upon being inaugurated to his second term after the contingent election.
On the other hand, Mr. Joe Biden is neither POTUS nor Acting POTUS. He is committing a Federal crime by impersonating the holder of a Federal office that he does not hold (because NOBODY holds it right now). All orders and appointments made by him are not worth the paper they are written on, he is essentially roleplaying. In fact, if Biden makes a "law" or "order" that Trump would like to keep (for example lowers the taxes or increases Vetrans' benefits), Trump has to copy them word by word and sign them himself, as legally, any actions of Biden have no legal power, no matter whether they are good or bad.
Due to the significance of the fraud and its consequences, Biden should be liable for treason. What he is doing is a malicious intelligence operation that is usually done to a country by its adversaries. After all, he manipulated and thus nullified a Federal election and is now impersonating the highest office in the country, misleading the public about its current holder. There is no President at all right now, and yet Biden claims to be President.
As people believe these lies, they follow "laws" and "orders" that do not exist. Nobody is legally obliged to submit to a "law" or "order" that has been "signed" by Mr. Joe Biden. In fact, if said "laws", "orders" or "obligations" directly conflict with currently legitimate law (anything passed by President Trump before January 20th and by Acting President Trump after January 20th), following them could be a crime (and another charge is added to Biden's indictment, he would be inciting crimes).
So let's imagine Trump banned the wearing of blue hats, and Biden "signs" a "law" that makes wearing blue hats not only legal but obligatory on certain days. He is essentially calling people to commit a crime, and anybody wearing a blue hat still commits a crime, as the "law" signed by Biden is void.
TL;DR: Neither Trump nor Biden are POTUS right now, the office of POTUS is currently unoccupied because there was no legitimate election and yet the terms ended. However, Trump automatically inherited not the office but the powers associated with it on January 20th from himself and became Acting POTUS.
Not claiming to be the first, but it was an obscure fork. It may be impossible to ever find out, but in all the tumult between the 6th and the 20th of January, at one point I thought that Pence resigned early, allowing Trump to appoint Pompeo as VP. (Technically he could do an interim appointment of anyone to any vacancy after the previous Senate session ended. Actually, using a resignation of the SOS and a secret interim appointment, even Richard Grinnell could be president now.) But that would be pointless unless he could be both VP and Scty of State at the same time. I still like the Pompeo is President theory, but unfortunately it's pointless to speculate when it's still a secret now, and the truth will eventually be public.
The terms "regent" or "successor" are not a part of the US's political language, so while I understand what you mean, I'm sure he won't have used those terms.