There is one structure that many linguists use but is seldom referred by grammarians in either English or Spanish: the middle voice.
You know the active voice:
María vende pan. | Mary sells bread.
and the passive voice:
El pan es vendido (por María). | Bread is sold (by Mary).
The passive voice can have an explicit agent (“by Mary”) or a tacit agent: “Bread is sold (period).“ The middle voice can be thought of as a passive voice without any explicit or tacit agent:
El pan se vende. | Bread sells.
In English the middle voice is very uncommon, few verbs admit middle voice and there is no grammatical mark from an active voice: “Mary sells” v/s “bread sells”. OTOH, middle voice is common in Spanish and looks grammatically as reflexive. Note that in
El pan se vende. | Bread sells. / Bread sells itself.
It is not bread the one making the sale, but what is sold.
The construction in
Yo me llamo Carlos.
is more similar to a middle voice. It does not mean that I call myself Carlos but rather that people call me Carlos, regardless of people actually calling me Carlos.
On the other hand, an expression as:
Yo me llamo por teléfono. | I call myself by phone.
is a really reflexive action.
Spanish language grammarians call «llamarse» a pronominal verb, rather than a reflexive verb.
Pronominal verbs use a reflexive pronoun, but do not have a reflexive meaning. Some times the pronominal verbs have a middle voice function, and sometimes it is a different meaning.
In English, the phrasal verbs sometimes have a complete different meaning than the meaning of its parts: “to make up” is not the combination of “to make” and “up”, so it cannot be translated as «hacer hacia arriba» or any similar expression in Spanish.
Yo me llamo Carlos.
shall not be translated, and shall not be though as meaning “I call myself Carlos.” It is translated as.
My name is Carlos.
Me llamo José, pero dime Pepe.
There are different ways to convey the meaning you expect of
My name is Joseph, but call me Joe.
Me llamo Joseph. / Mi nombre es Joseph. / Joseph dice la tarjeta/el pasaporte.
Pero dime Joe. / Pero soy Joe para los amigos. / Pero llámame Joe.
Of course, you want the contrast between llamar and llamarse:
Me llamo Joseph, pero llámame Joe.
The first construction is:
[(tacit) nominal pronoun] + [pronominal verb _llamarse_ in indicative] + [the name]
(Yo) me llamo Carlos., (Ella) se llama María., (Tú) te llamas Joseph.
The second construction is (in imperative):
[transitive verb _llamar_ in imperative] + [enclitic pronoun of direct object] + [the name]
Llámeme Carlitos., Llamadla Marucha., Llámate Pepe..
(Note: «Llama»/«llámame» is the imperative for «usted», while «llamad/llamadla» is imperative for «vosotros»)
If you don't want an imperative but rather an indicative sentence:
Se llama Joseph, pero lo llaman Joe. | His name is Joseph, but they call him Joe.
Also, the first is the pronominal verb llamarse, while the second one is the transitive verb llamar.
In most of these examples, I would rather use the verb decir for the second part.
It might not make sense to you if you translate «decir» as “to say”/“to tell”, but another meaning of «decir» is “to call smthg as smhw”, particularly when used with an enclitic pronoun.
— ¿Cómo se le dice a donde se dobla el brazo?
— Codo.
— How do you call where the arm bends?
— Elbow.
The best way to say what you want to say:
My name is Joseph, but call me Joe.
Me llamo Joseph, pero dime Joe.