Why does the number of extrasolar planets vary in different sources? There are at least three catalogues that are counting up-to-date discovered exoplanets. What criteria do they use to count the number of planets? Every catalogue has a different count value.

So at this moment various catalogues as:


exoplanet.eu
Open exoplanet catalogue
NASA exoplanet archive
etc.


show values of confirmed number of exoplanets between approx. 3500 and 4050.

Why do they disagree with each other?
As of April 6, 2020: 


exoplanet.eu shows 4241 planets;
the Open Exoplanet Catalog shows 3852 planets of which 3504 are confirmed;
the Nasa Exoplanet Archive shows 4144 confirmed planets;
exoplanets.org indicates a total of 5747 planets of which 3262 are confirmed planets;
...


The number of confirmed planets are different from one website to another simply because what exactly is being counted is different. Each one of these databases has a different criteria for deciding whether an exoplanet is going to be counted or not. To understand the differences between these numbers, you have to ask


How is an exoplanet defined?
How solid must the evidence be for the exoplanet to be counted?
What detection methods are included?


The answers to these questions are usually in the methodology section of the website, or in a dedicated article. 

Here are a few examples of differences:


exoplanets.org uses a cutoff of 24 Jupiter masses for what constitutes a planet or a brown dwarf, whereas the NASA Exoplanet archive uses 30 Jupiter masses. I haven't found a methodology section for exoplanet.eu, so I emailed the people maintaining the database, and they indicated they use a 60 Jupiter mass cutoff. 
exoplanets.org only shows planets that appear in peer-reviewed publications with well-measured orbital parameters, rather than giving a comprehensive list of all planet announcements.
the Open Exoplanet Catalog is decentralized and community driven, so on different instances of the database, you might have different numbers (which the authors consider to be a feature, not a bug).


In addition to this, each of these websites might also be updated at different times. For instance, the NASA Expolanet Archive is updated once a week, but exoplanet.eu is updated daily. But other websites might be updated at different times and with different frequencies. Because of this, two websites with the same criteria might still show different numbers.