I am not very strong on statistics and I am having problem in deciding how to treat my data. I collected production data showing speakers a video and asking them to tell to another participant what they just watched. And I did this for Turkish and Dutch speakers. Each participant (20 per language group) produced several sentences but not equal amount of sentences. I look at subject arguments only (e.g. in a sentence like "the woman took a jar", 'the woman' is the subject argument). Now I want to know whether Dutch speakers are more likely to use pronouns e.g. 'he', 'she', 'they' but all these collapsed as one category, pronouns (my dependent variable) than Turkish speakers. Sometimes speakers used a pronoun, sometimes a noun e.g. the woman. But note that this was a free production data, and later on I grouped the utterances as noun and pronoun. What I want to know is whether I code the data in binary fashion (1 = presence of pronoun; 0 = absence of pronoun) or whether I take proportion of pronouns out of all subject arguments (= number of sentence) make a theoretical difference. I know the type of analysis I should use will differ according to the type of the dependent variable (proportion or binary), but I want to know whether one way of coding the data is better than the other. I should note that if I code the data in a binary fashion, I will have different number of data points across participants because each participant used different number of sentences and also different number of data points across the two language groups. If I go for the binary coding and perform binominal logistic regression, is it a problem then that I have different number of data points per participant? If I take proportion of pronouns, I will have one data point per participant and 20 data points per language group.
Try examining each sentence individually, marking it as a $1$ if a pronoun is used, and $0$ if not.
Then if a subject speaks $3$ sentences, where he uses a pronoun in two of them, give him a score of $(1 + 1) / 3$. (Divide by the number of sentences spoken.)
You might also be interested in seeing if there is a statistical significance to the number of sentences spoken by each language user.
I will also say, however, that your first approach makes perfect sense. If a subject uses a pronoun at all, give him a $1$, otherwise $0$.