This is not so much a question of semantics (meaning of words), rather of pragmatics (use of words).
In German the word Führer is compromised for obvious reasons (look out for it in the 1933-1945 era). So, usually today one would avoid using the word Führer except when indeed the Nazi dictator is meant.
Exception are a number of other uses of Führer, usually in composita. Depending on context, Führer can mean (or be used in)
a book (Reiseführer, Pflanzenführer)
a person leading a group of tourists, e.g. in a national park or so, or in a museum: Fremdenführer, Museumsführer, Stadtführer
when speaking of operating a vehicle or machine: Fahrzeugführer, Kranführer, Baggerführer, Lokomotivführer, Maschinenführer, and so on, and most famously of course Führerschein (driving licence).
Perhaps some other rarer cases which do not come to my mind at the moment
In such cases the proper word would be the full compositum (Kranführer) but when the context (construction site with a crane) is clear, one might say, for brevity, der Führer verließ den Kran und holte sich ein Bier, without risking being misunderstood.
Anführer though is a very broad term, used for all types of "leading positions" in a group of people when speaking about them informally. You would rather not use it in cases where a formal title for that position is commonly accepted. E.g. you would not speak of der Anführer des Rathauses when you mean the mayor (unless you want to be ironic or sarcastic). You would probably speak of der Anführer der Clique (youngsters sticking habitually together), also der Anführer der Räuberbande. However, Anführer is not necessarily bad. You may say die Anführerin der Klimaschutz-Bewegung und mean it positively. The word Anführer/in is then more about the informal character of the leadership position, not about your positive or negative attitude towards the topic.
Back to your Merkel example: Reasonable authors would not call Mrs Merkel Führerin of anything right because of the association of the term with the Nazi dictator (or generally with authoritarianism, fascism, totalitarianism).
Perhaps you like the popular joking verse
Der Führer ist ein armes Schwein,
denn er hat keinen Führerschein.
which of course refers to A. Hitler.