Arabic meaning of the crucial words
First of all we need to clarify a few meanings of Arabic words:
The meaning of the noun:
which actually refers to obscene, dirty and vulgar language etc..
refers to being excessive, heavy, gross, unreasonable or dirty, filthy, obscene and vulgar (depending on the context).
For example in Arabic one could say: A person is tall in a fahish manner (meaning that he is tall in an excessive manner)
If you found other meanings they also may apply basically both the noun and verb refer to anything which people consider as bad or they would reject.
From the same root you may find the noun:
الفحشاء or الفاحِشة
Which depending on the context refers to atrocity, monstrosity, abomination (See for example 3:135 and 4:22) or adultery, fornication and whoredom (see for example 4:15). Note that in the qur'an the second meaning is always meant in a clear context to a sexual act like zina or homosexual relationships and you will usally find them explictly quoted in the same verse (or in the surroundings), while the first meaning mostly is mentioned in connection with منكر (usually translated immorality and wrongdoing).
So basically a relation to zina and sexual bad words is not necessarily given, but it also can't be excluded.
A look into hadith interpretation
As for the meaning in the hadith in 'Omdat al-Qari عمدة القاري (see here in Arabic) you will find the following interpretation of the words from the hadith in Sahih al-Bukhari:
قوله: " لم يكن النبي صلى الله عليه وسلم فاحشا " من الفحش، وأصله الزيادة بالخروج عن الحد.
as for "The Prophet (ﷺ) never used bad language neither a "Fahish" from الفحش and it comes from the exaggeration and excursiveness.
قوله: " ولا متفحشا "، أي: ولا متكلفا في الفحش،
as "nor Mutafahish" it means a person presenting himself (or acting) in the manner of الفحش (with exaggeration).
حاصله أنه لم يكن الفحش له لا جبليا ولا كسبيا
In summary he neither was into الفحش by nature nor by acting similar to (or learning) others.
And imam al-'Ayni also quoted other related narrations such as this from Jami' at-Tirmidhi and this from Sahih Muslim. In his commentary on another occurrence of this hadith imam al-'Ayni also pointed at the fact that فاحش applies to anything that one can't change as it is given us by Allah (or by birth or nature as some people may say). While متفحش refers to any (bad) attitude that one could learn.
So what we may learn from this is that our Prophet () was never born as a person who uses bad language nor has ever applied it in his life.
My own conclusion is that bad language refers to saying words that may hurt to other people it -at least in first place- doesn't refer to sexual bad words: It's more something like "idiot!" etc.
Here some further ahadith references which were quoted in this fatwa islamqa #145395 (in Arabic) related to the context of your first hadith and people with an insolent tongue:
In Sahih Musilm and Sahih al-Bukahri, also in Sahih al-Bukhari, in Sunan abi Dawod and in Jami' at-Tirmidhi (a narration that might refer to the extended meaning with sexual content)