Well, yes and no. Scottish Gaelic and Manx are really direct descendants of Old Irish, not some 'proto-Gaelic language'.

As MoR pointed out, Irish is indeed referred to as 'Irish' when speaking English. In Irish, it is called Gaeilge (i.e. Gaelic) not Éireannach (i.e. Irish).

The same is true for the others. Scottish Gaelic-speakers call their language Gàidhlig (Gaelic) and Manx-speakers call their 'Gaelg' or 'Gailck'.