There are several juridical nouns for different kinds of homicides. @KilianForth mentioned the most important ones in a comment already: Mord ‘murder’, Totschlag ‘man-slaughter’, fahrlässige Tötung ‘involuntary homicide’, Körperverletzung / Gewaltanwendung / unterlassene Hilfeleistung mit Todesfolge.
There are also several (related) verbs: ermorden, töten, erschlagen and more specific ones.
Although it would be possible to derive actor nouns from these, e.g. Töter, most are not conventionalized.
I think Totschläger in particular used to be more common, but the word is more likely being used to refer to a certain kind of club nowadays.
Note that Totschlag does not require beating or hitting, but erschlagen usually does.
Totmacher, like tot machen ‘make dead’, is only found in child-like speak.
In legal contexts, Mörder is restricted to intentional killings and some further conditions, otherwise it’s used frequently and without much discrimination.
The anglicism Killer (and the verb killen) tends to be limited to the more ruthless cases, as in Auftragskiller ‘hitman’.
There is a number of special nouns by frequency, method or target of killing, but most are just compounds of Mörder: Mehrfach- / Massen- / Serien- / Kinder- / Frauen- / Sex(ual)- / Ritual- / Raub- / Axt- / Giftmörder, but also Würger ‘strangler’ (and erwürgen), Todesschütze ‘shooter’, Attentäter ‘assassin, terrorist’, Messerstecher ‘stabber’.
A killer who hasn’t been convicted yet is usually called mutmaßlicher Mörder or vermutlicher Mörder or, if one wanted to express some doubt about the verdict, angeblicher Mörder.
These adjectives can, of course, also be applied to the more generic Täter, which requires contextual information about the crime in question.
You may want to note that several verbs relevant in this context require the object Mord (or Totschlag etc.) to be in genitive case, which is rather unusual, e.g. jemanden des Mordes beschuldigen / verdächtigen and jemanden wegen Mordes anklagen.
There have actually been court cases to determine whether the Tucholsky quote “Soldaten sind Mörder” was a defamation of soldiers. Better avoid. “Soldaten töten” would be much less controversial.
There are few other “jobs” that involve killing of humans.
Executioners are properly called Henker or Scharfrichter, but if there still was capital punishment in German speaking countries I guess there would be a more complex, harmless sounding, bureaucratic designation like “ausführender Justizvollzugsbeamter”.
Suizidhelfer would be used for assisted suicide, whereas any compounds of Selbstmord are avoided for the most part.
The (yellow) press features a whole bunch of (mostly awful) metaphors, e.g. Todesengel or Todesfee ‘angel / fairy of death’ for ICU nurses pulling the plug or overdosing their patients.