How is the differential definition of money supply, velocity, and GDP defined?

That is not a differential form of the equation. $Delta$ there stands just for changes not differential. You can derive the equation in such form by log-linearization.

Take the original equation:

$$MV=YP$$

Apply natural log to both sides:

$$ ln M + ln V = ln Y + ln P$$

And now you are done, since in an equation that is in log-log form you can interpret everything in terms of changes.

Hence:

$$ ln M + ln V = ln Y + ln P implies Delta M + Delta V = Delta Y + Delta P$$

In regard to the question about the meaning of $Delta$ notation I agree substantially with the answer given by @1muflon1. However I could only become convinced of this answer by expanding the "logic" implied in the notation with reference to the information below.

Notation Used in the Article (See Question)

https://thismatter.com/money/banking/money-growth-money-velocity-inflation.htm#:%7E:text=The%20classical%20theory%20of%20inflation%20states%20that%20money%20growth%20causes,services%20per%20unit%20of%20time.

"Equation of Exchange"

$MV = YP$

"It can also be derived from the equation of exchange, for small changes in the factors

$Delta M + Delta V + Delta P + Delta Y$

The Quantity Theory of Money (1 page)

https://files.stlouisfed.org/files/htdocs/publications/es/06/ES0625.pdf

The quantity theory of money (QTM) asserts that aggregate prices (P) and total money supply (M) are related according to the equation P = VM/Y, where Y is real output and V is velocity of money. With lower-case letters denoting percentage changes (growth rates), the QTM can be expressed as p = v + m – y, with p as the rate of inflation and y, v, and m as growth rates of output, velocity, and money stock, respectively.

This reference expresses QTM (Equation of Exchange) in terms of percent change but without any mention of the reasoning, derivation, use of natural logarithms, or the method for computing percent change.

Use of Logarithms in Economics

https://econbrowser.com/archives/2014/02/use-of-logarithms-in-economics

Thus if you earn r% interest that is compounded continuously, at the end of the year your money will have grown by $P_{2}/P_{1}=e^{r}$. For example, if r = 4%, continuous compounding would actually give you 4.08% at the end of the year, a little more than the 4.06% from quarterly compounding or 4.0% with no compounding.

Taking natural logarithms is just the inverse of the above operation: $ln(P_{2}/P_{1})=r$, or since the log of a ratio is the difference of the logs, $ln(P_{2})-ln(P_{1})=r$. In other words, taking the difference between the log of a stock price in year 2 and the log of the price in year 1 is just calculating a rate of return on the holding, quoted in terms of a continuously compounded rate.

For low values of r, the continuously compounded return is almost the same as the non-compounded return, so that the log difference is almost the same number as the percentage change. For the example just given, the percentage change is $(P_{2}-P_{1})/P_{1}=0.0408$ whereas the log change is $ln(P_{2})-ln(P_{1})=0.04$. So any time that you see a graph that is measured in logs, an increase of 0.01 on that scale corresponds very closely to a 1% increase. A graph that is a straight line over time when plotted in logs corresponds to growth at a constant percentage rate each year.

A Monetarist Model of the Inflationary Process (11 pages)

https://www.richmondfed.org/~/media/richmondfedorg/publications/research/economic_review/1975/pdf/er610602.pdf

Page 2 (Nominal page 14, final paragraph)

Three other features of the model should be mentioned at the outset. First, all relations are linear and are expressed in logarithmic form. There is a specific reason for this formulation. Modern monetarist analysis is usually stated in terms of percentage rates of change of the relevant variables. And since the percentage change of any variable over a given interval of time can be represented mathematically by the first time difference of its logarithm, it follows that a log-linear formulation facilitates the analysis.

Page 3 (Nominal page 15, second full paragraph)

As for notation, the model employs the following symbols. Let m, be the money stock, y actual real income, ... and p the price level - with all variables expressed as logarithms.

The symbols $Delta$ and $Delta^2$ appearing before a variable denote first and second time differences, respectively, so that the model is effectively expressed in terms of proportional rates of change and rates of acceleration or deceleration of those rates of change.