Yes, this is perhaps why it's so hard to get anywhere when using the term ethnic, even if it's applicable.
To make this more complicated, there are no clear definitions of race or culture. As a sociologist/biologist, one learn the limits of those terms. Genetics have learned us that terms like "species" can be misleading and confusing. If you travel by car through the world, you will notice that the swift from one race to another isn't clear.
I am both, and I see it from both arenas.
The true definition in the biology field of ethnicity is used to classify "differences between humans". Differences between humans as where they are identified by people through DNA nucleotides.
See an example study from Stanford
here where the word ethnicity is explicitly used to define biological differences between humans. In this particular case, it makes mention to G6PD, glucose-6-phosphate-dehydrogenase, which is an enzyme responsible for ATP and energy synthesis in cells as it travels through the bloodstream. The particular article focuses on a deficiency that explicitly affects certain 'ethnicities', which in this case, are the Italian blood-related persons because their body does not produce enough of it compared to a human from another region. These are just a few of the problems you'll find out from time to time that affects a person from one region moreso than another, because of a genetic issue.
Socially, and in some cases sociologically, ethnicity is generally defined by the culture in that group, because the types of humans (as in color, or race, whichever you call it) that share something in common. Which is why you have some people who are thinking ethnicity just means the culture, and pride to it. True american patriots feel united through the 4th of July, for the same reasons olympic gold medalists feel a sense of relatedness when their national anthem plays after their win.
Yet, during the 4th of july, blacks and whites can equally feel a sense of patriotism even though it applies to the "white colonists" more specifically. Asians and hispanics are less likely to feel that sense of union, simply because they don't have the heritage associated with it. Which in a boundary like this, it applies to the social realm.
In the legal world today, the biological definition almost always takes over the social definition. I'm pretty sure you won't find any hospital willing to change it on their forms. In the social world, whatever goes, no one is going to judge you based on what you personally feel aligned with. It has less prevalence than the bio definition when we are not speaking in legal terms, so that is why whatever goes in that arena.
Language is not a key factor, as there
many ethnic groups that consider themselves different from another group, yet speak the same language. Check out the ethnic groups on wiki for some examples.