An ion of chlorine with a charge of −1 is called a chloride ion/anion.
Are there terms for other ions of chlorine, for example, $\ce{Cl+}$ or $\ce{Cl^2-}?$
An ion of chlorine with a charge of −1 is called a chloride ion/anion.
Are there terms for other ions of chlorine, for example, $\ce{Cl+}$ or $\ce{Cl^2-}?$
Assuming that "term" in your question refers to chemical nomenclature, the authoritative source of information would be the current edition of IUPAC Red Book [1]. From [1, p. 70]:
IR-5.3.2.2 Monoatomic cations
The name of a monoatomic cation is that of the element with an appropriate charge number appended in parentheses.
[…]
- $\ce{I+}$ iodine(1+)
Analogously, the name of $\ce{Cl+}$ is chlorine(1+).
Further from [1, pp. 72–73]:
IR-5.3.3.2 Monoatomic anions
The name of a monoatomic anion is the element name (Table I) modified so as to carry the anion designator ‘ide’, either formed by replacing the ending of the element name (‘en’, ‘ese’, ‘ic’, ‘ine’, ‘ium’, ‘ogen’, ‘on’, ‘orus’, ‘um’, ‘ur’, ‘y’ or ‘ygen’) by ‘ide’ or by directly adding ‘ide’ as an ending to the element name.
Examples:
- chlorine, chloride
[…]
Charge numbers and radical dots may be added as appropriate to specify anions fully.
Examples:
14. $\ce{O^2-}$ oxide(2−), or oxide
So, the name for $\ce{Cl^2-}$ is chloride(2−).