Well, it depends who you were and what context you were in -- you might say "Austria", "Bohemia/Moravia/Silesia," or what region or town you were from depending on who was asking and why, but obviously this is also true of people today so this doesn't answer your question. It seems like you are asking about national versus imperial identity. In 1910, many people in the Bohemian lands would have considered themselves both members of the Czech nation and citizens of Austria, without seeing a contradiction between these identities at all. They might have been Czech nationalists and advocated for more power and autonomy for Czechs within the Austrian empire, or they might have been radical Czech-speaking workers who supported social democracy for both the nation and the working class, or they might have been just ordinary people who liked reading Czech newspapers and magazines and this developed their sense of Czech patriotism and belonging, and this would not at all suggest that they didn't also consider themselves to be members of the Austrian polity.

Some people would not have had a very strong national identity, or they might have called themselves a Czech in some situations and a German in others. Since there were many bilingual people (especially in particular regions), nationality could be a matter of affiliation -- you joined the Czech hiking group, read Czech newspapers, and sent your kids to the Czech elementary school, so you were Czech, even if you spoke both Czech and German equally well, and maybe your mother really never spoke much Czech and your brother was in the German hiking group. This was a time when both Czech and German national movements were very concerned with winning as many people as possible for their respective nations, so you might find yourself being denounced as a national traitor in the local newspaper if your neighbor judged that you were a Czech who was sending you children to a German-language school. If you lived in Moravia, the Czech nationalist activists might actually come and "reclaim" your kid from a German school if it could be proven in court that they were "actually Czech." Although nationalists often scorned people who didn't affiliate themselves one way or the other as either ignorant and backwards or untrustworthy and dishonest, the phenomenon of "national indifference" continued well into the twentieth century. Still, by 1910, both the Czech and German national movements had firmly established themselves among the population.