THANK YOU ALL for your advice, hints and ideas. With your help I have now I solution, that works for me!
I call the result: "How to save a series of pictures in safari, when every picture is on a new site"
I'll give you the details now:
The solution consists of two macros and is tested only in the safari browser.
First of all: It is advisable to go to preferences of the FINDER and to checkmark "show all suffixes" (I don't know how this is written exactly, since I'm working on a German Mac. In German its "Alle Dateinamensuffixe einblenden". When this preference is made, you don't have to bother about the format of the image.
And it is a clever idea too, to go in the preferences of SAFARI and to choose in the common section for the value of "place to save" ("Speicherort für Downloads") the value "ask every time" ("Jedesmal nachfragen").
Now the macros. The first one is simply to initialize for a new series of pictures, that I want to save.
The first command initializes a global variable (it has to be global, because the action is repeated every time from the beginning).
Then I have added just an alert, to let the world know, that the variable is initialized and has a value of 1.
Before using the second, the main macro, you have to manually navigate your browser to the first picture you want to save, and to save it in the directory (that you probably create during this save) and to save this first picture as "001,jpg" (or 001.tif or 001.png - depending on your pictures). Then you navigate to the second picture of the series and hover the mouse over the picture.
NOW you can start the second macro. Here is it:
"Move and right click" opens the context menu, normally shown when clicking the right mouse button.
"Insert text" is the exact text shown in the context menu. In German "Bild sichern unter ..." in English "Save image as ..."
The simulated "return" opens the saving dialog window. Fortunately the focus is already on the field of the filename, so the desired filename can be written with another "Insert text" - but before that the variable "GVar__MyCounter" is incremented by 1 AND with the additional option to format the result the 000-group gives leading zeros if needed.
After the already mentioned Insert text another return is simulated to close the file dialog window.
Last not least the user should know, that the action is terminated. This could be done with an alert. But since often the pictures in the browser are made as links, where a click leads to the next picture, I have simulated a left mouse click instead.
Since the opening and closing of the file dialog is time consuming the whole macro is saturated with pauses.
I hope, that someone finds this useful. And again: Thanks to all