It depends. "Old" is always relative to what stages of a language are attested. We usually use "Old" to describe the first stage where a language becomes well attested.

The categorization of English into three periods, Old/Middle/Modern, is convenient largely due to the mutual unintelligibility of those three stages. If we go back further, we can reconstruct Primitive or Proto-English. Even further and you have Proto-Germanic. Go back even further than that and you have Proto-Indo-European.

Other languages can be broken up into similar Old/Middle/Modern stages when it's convenient to do so. We have Old Irish, Middle Irish, and Modern Irish. There is some Primitive Irish attested in Ogham inscriptions, but not much. The Old stage is where it we first start seeing large amounts of Irish literature. We also have Old Welsh, Middle Welsh, and Modern Welsh. Old Welsh isn't as well attested as Old Irish, but there's enough of it. With French, we see Gallo-Romance dialects of Vulgar Latin become Old French. Then there's Middle and Modern French.

Languages that are less well attested through time can't be categorized like this. Navajo, for example, wasn't recorded at all until the 19th century. Previous stages of the language existed, of course, and the ancestor language called Proto-Athabascan can be reconstructed to an extent. The only Navajo we have is "Modern" Navajo. There are two languages called Tocharian that are attested in translations of Buddhist texts. We call them Tocharian A and Tocharian B. Because we don't have earlier attestations for either, there is no Old Tocharian. And because there are no speakers of a descendant language, we can't speak of a Modern Tocharian.

Sumerian was in use for a very long time and the Old/Middle/Modern paradigm doesn't fit it so neatly. Instead, it's broken up into Archaic, Old, Neo, Late, and Post-Sumerian.

For a language that has been revived, like Cornish, things get messy. We have Old Cornish and Middle Cornish attested in medieval manuscripts. Because the language has been revived, we need to make a distinction between Modern Cornish and Late Cornish. Late Cornish is the last stage of the language as spoken by native speakers prior to the revival. Modern Cornish not only refers to the language today, but specifically to one version of it. Sometimes the term "Revived Cornish" is used more generally.