Are there terms for ions of chlorine other than chloride?
Assuming that &quot;term&quot; 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−).
Reference

IUPAC. Nomenclature of Inorganic Chemistry, IUPAC Recommendations 2005 (the “Red Book”), 1st ed.; Connelly, N. G., Damhus, T., Hartshorn, R. M., Hutton, A. T., Eds.; RSC Publishing: Cambridge, UK, 2005. ISBN 978-0-85404-438-2.