Keller's Learn to Read Latin says on p42
In the sentence "The farmer came to the party with a poet", the phrase "with a poet" indicates that the farmer was accompanied by a poet. The phrase "with a poet" would be expressed in Latin by the preposition cum (with) and "poet" in the ablative case; the syntax of that word would be Ablative of Accompaniment. NOTE THAT THE ABLATIVE of ACCOMPANIMENT REQUIRES THE PREPOSITION cum.
In the sentence "The farmer is fighting with a sword," the phrase "with a sword" indicates what the farmer uses to fight, or the instrument by means of which he is fighting. The phrase "with a sword" would be expressed in Latin by the word "sword" in the ablative case with no preposition; the syntax of that word would be Ablative of Means (or Ablative of Instrument). NOTE THAT NO PREPOSITION IS USED WITH THE Ablative of MEANS.
and on p83
126. Ablative of Manner
A noun in the ablative case may express the way or manner in which an action is performed. An ablative so used is called the Ablative of Manner. WHEN A NOUN FUNCTIONING AS AN ABLATIVE or MANNER is NOT MODIFIED BY AN ADJECTIVE, THE PREPOSITION cum MUST BE USED. WHEN A NOUN IS MODIFIED BY AN ADJECIIVE, cum Is OPTIONAL. For example:
Agricolae cum studio laborabani.
The farmers with zeal were working.
The farmers were working with zeal.
The farmers were working zealously.
Magna (cum) cura in via ambulabo.
With great care in the street I shall walk.
I shall walk in the street with great care.
I shall walk in the street very carefully.
The syntax of each italicized word (studio, cura) is Ablative of Manner.
Is Ablative of Manner a kind of Ablative of Means/Instrument? What differences and relationships are between them?