What is meant by an inflation rate spoken in the typical news?

The news usually says that inflation for a certain month is say 2.57%. It is never said relative to what. Should I assume that is relative to the previous month?

Yes.

Ideally the news should specify whether it's month-on-month (MoM) or year-on-year (YoY).

But if it simply says inflation in July 2020 was $X$ (without specifying whether it's MoM or YoY), then it should mean MoM (i.e. relative to June 2020).

If that's so, then the more inflation rises, the more likely the percentage reported will be small and eventually negative --- otherwise, inflation would be rising exponentially.

I do not understand this sentence at all or what the logic is.

Wouldn't be more useful if we would look at inflation since the a certain historic mark? In the US, for example, shouldn't we look every day how much money the dollar has lost its value since 1917 or 1971?

There are such measures. For example, the US BLS Inflation Calculator tells us that $100 in Jul 2020 has the same buying power as $4.52 in Jan 1917 and $15.36 in Jan 1971.

Inflation is by definition change in average price level. If we denote price level as $P$ and inflation as $pi$, it is given by:

$$pi = frac{P_t-P_{t-1}}{P_t}$$

Hence yes it is given relative to its previous time period as its growth rate.

You can look at inflation since any date you want since $t$ and $t-1$ can be any dates. However, the most relevant is the current rate of inflation as just from purely practical perspective for conducting business or policy its current or future expected rate of inflation that matters most since past is already past and people make economic decisions in present or plan for future.

In media the inflation number that you see is generally the Year-on-Year inflation rate, i.e.,

$$frac{I_t-I_{t-12}}{I_{t-12}}$$

where $I_t$ is the price index (CPI or WPI). For most countries, the index is a fixed base index. So the absolute value of the index gives you the idea of where the prices are relative to the base year.

Inflation rate on the other hand gives you idea of how much the index has increased as compared to same month previous year. The main reason for using YoY over MoM is to prevent seasonality misleading you.

For example, if media reported MoM, every December you'd see high inflation because of holiday season. So inflation of December is not strictly comparable with other months because of inherent seasonality. To prevent this YoY is used.

On your second point: No inflation does not keep decreasing with time. This would be true if price increases with a decreasing slope. There is no economic reason to believe that.

On third point: if we only looked at inflation rate from a historic point then that number would rather be an increasing number. Then again you would not be able to compare it with previous months' or years' numbers as then the difference of inflation would be of interest.