As I understand it, while "marcher" means "to walk", "promener" is used as in "to go for a walk". So, in this case, marche and promène make the difference between "I walk my dog" and "I take my dog for a walk".
Hopefully someone fluent in French can point us in the right direcction.
Think of the verb "promener" to mean "to take (_) for a walk" or "to walk _" where the blank space represents a person/thing. So when using the verb promener you always need to include a direct object, i.e. the thing that is being walked. Only pronouns (me, te, le, etc.) go before the verb as in "Je me promène" ---> "I go for a walk" or more literally "I take myself for a walk" or "I walk myself." You don't need to have an object before the verb if the thing you're walking isn't being referred to with a pronoun like this example "Je promène mon chien" because "mon chien" is the direct object (what is being walked) and it isn't being referred to using a pronoun so it goes after the verb.
There is an English word, "promenade" which means a certain type of walk verb [ no obj. ] take a leisurely walk, ride, or drive in public, esp. to meet or be seen by others: women who promenaded in the Bois de Boulogne. • [ with obj. ] take such a walk through (a place): people began to promenade the streets. • [ with obj. ] dated escort (someone) about a place, esp. so as to be seen by others: the governor of Utah promenades the daughter of the Maryland governor.
So, to check if this was an acceptable usage in Francais, I threw myself under the bus with "I promenade my dog."
Shall I report this or shall I assume it to mean there is no French equivalent for the English "promenade" (though please note the reference to "Bois be Boulonge" in the definition) Irony, n'est pas?
I think you are correct that there is no French equivalent. I note that Larousse does not suggest "promenade" as an English translation for "promener", although there is a bit of that notion in definition 4 (which should be noted, is considered "soutenu" - formal or old-fashioned)
Even more to the point, when translating in the other direction, Larousse gives only "(se) promener" for the English verb "promenade", but notes that the usage is "(formal & humorous)"
What does "I walk my dog" really mean in English? A little bit confusing sentence for me :)
"Se promener" really just means to go for a walk. I don't believe it carries the showing-off connotation in French.
According to Larousse, if you want to translate the transitive version of "to parade", you'd say "faire étalage de" (to display or flaunt)
The French verb 'promener' was used often in the 19th century in the USA. It was used to denote a walk along public avenues in one's finest apparel. That walk was called a promenade by the papers of the day. Clearly the French use the verb to take a walk or to walk someone/something. (Now what did Debussy mean by 'Goliwog's Cakewalk?).
The way I think of this distinction is in Spanish: Marcher is akin to "caminar" or to walk, which refers to the mechanical action of putting one foot in front of the other Promener on the other hand is akin to "pasear" or to take a walk, which refers to taking a leisurely walk, for fun and (maybe) sightseeing, either by yourself or with a companion.