This question is full of misconceptions, I think those have to be corrected first.
Unlike a normal company, the government can't simply go to the bank and ask to borrow existing money from the bank, rather (it is my understanding) that the government goes to the treasury and asks for more money to be printed so that the government can borrow from the treasury, to which the treasury always responds yes.
No. First US treasury is literally part of the executive branch of the US government - thus US treasury is the government even in the most strictest and narrow sense of the word. Second, printing in US is done by Bureau of Engraving and Printing, which is overseen by the treasury but it is not the treasury, and minting is done by US mint. Third, US government can borrow money from bank or to be precise from a central bank, in the US that would be the Federal Reserve (Fed). That is actually where US is borrowing most of its covid stimulus money (as explained in this article by Committee for a Responsible Federal Budget). Moreover, most of the money nowadays is not printed but it is created electronically by Fed - there is no meaningful economic difference between printing a note and creating an equivalent electronically but still its good to make that distinction to know where the money is coming from (i.e. not really from Bureau of Engraving and Printing but mostly from Fed).
It makes sense to me that if America wanted to borrow money from China, the only money that it could borrow would be RMB, because the only money that China can print is RMB
No, if China wants to buy US government debt it has to do it in dollars. The way how it is done is that China first converts its RMB to US dollars on foreign exchange and then pays for US treasury bill in dollars. Of course, nowadays the sale might be set up in a way where everything is done in one step, but US debt is denominated in US dollars. When US government issues its debt it gets dollars and it promises to pay back dollars.
Answering the Main Question:
Why is it that when a government prints a ton of money, it causes inflation, rising of prices, and so on as in Zimbabwe many years ago, but when a government borrows the same amount of money from another country, it does not cause inflation at the same level?
Because, while virtually all conventional economists agree that there is positive relationship between increase in money supply and inflation, the relationship is not proportional and there are multiple other variables that play a role.
The simplest model to describe money market equilibrium (which determines price level and thus inflation) is given by equation of exchange (See Mankiw Macroeconomics pp 87) as:
Where $M$ is the money supply, $V$ velocity of money, $P$ price level and $Y$ output. Solving for price level and log-linearizing we get:
$$\ln P=\ln M+\ln V−\ln Y$$
Thanks to the logs the above equation can be interpreted in terms of changes, hence for example if output $Y$ increases by $1\%$ price level would be expected to decrease by $1\%$ and so on. In turn changes in price level are by definition inflation (or deflation for negative changes) so setting up the equation this way allows us to directly talk about inflation.
The equation above showcases that its not only money supply that matters for determination of price level, but also output and more importantly velocity of money. It is actually the last one (velocity of money) that is argued to be culprit for why US is not experiencing much inflation and indeed you can see from the data that velocity of money in recent times in US dropped to its lowest levels it ever was (see the data provided by Fed here). The velocity just between end of 2019 and present dropped by over $35\%$, meaning that even if money supply would increase by $35\%$ there would be no inflation given the model above.
Next, when I introduced the model above I mentioned it is most simple model (I did not wanted to overwhelm you by more complex model since you are clearly not an economist). In more complex models what matters is not just the variables above but also their expectations and how are people's expectations of inflation anchored (see Romer Advanced Macroeconomics for overview of more nuanced models). It is also being argued that peoples inflation expectations are low due to Fed pursuing policy of low inflation for very long time. The job of modern independent central bank was for very long time at least partially to maintain price stability which for long time meant fighting inflation so it is very hard for central bank to be credible when it tries to increase inflation.
Furthermore, the increase in money supply in US is nowhere near the amount that occurs in countries suffering hyperinflation. In the US the money supply (measured as M2) increased since last year by approximately $22.5\%$ according Fed data, whereas for comparison in Venezuela that is experiencing hyperinflation the money supply expanded by about 753% (according to data by trading economics). US is not even close to that mind boggling rate of money supply increase so there is no reason to even expect hyperinflation (which is typically inflation that is at least $50\%$ per month).