I've heard some experts say 1% and other experts say 50% to describe this same scenario. Can both be correct? Which one is more mathematically correct? How do you remove ambiguity when trying to describe this scenario?
It is also very common to say (in this specific scenario) that interest rates have increased by one percentage point. In other words a percentage point is an absolute difference of $0.01$ when the percentages are expressed as decimals.
If you say it went up by $1\%$ you mean it increased by an additional $1\%$ of the loan amount.
If you say it went up by $50\%$ you mean the interest rate increased by $50\%$ of its previous value.
Both percentages are correct, it is just that they refer to percentages of different quantities. Either way, in a case such as this where there is ambiguity, the quantity that the percentage refers to should be made explicit.
When the price of a head of lettuce goes from two dollars to three dollars, we either say it increased by one dollar, or we say it increased by fifty percent, and there's little risk of confusion. The problem here is that the thing that's increasing is itself a percentage. The best way to avoid ambiguity is to simply say at the outset that the interest rate went from two percent to three percent. If you then want to refer to it as a one-percent increase or as a fifty-percent increase, it should be clear to the audience what you mean.
You can say the interest rate increased 100 basis points, that is how interest rate changes are usually characterized. 100 basis points is equivalent to 1%. So you can say increased 1%.
The going from 2% to 3%, the rate of interest increased by 50%.
So "increased" and "increased by" are the operative terms.
Anyway, unless you live in Zimbabwe, either way, everyone will know what you are talking about. A 1% change in the rate of interest from x% to 1.01x% would be contextually distinct from a 50% increase in interest rates (as we have recently seen from say 2% to almost 3%).