You normally use $\theta_i$ to indicate that the probability of "success" depends on each observation, i.e. it is not constant across your sample. Logistic regression is a famous example of such practice. There you model each probability in terms of one or more explanatory variables:

$$\theta_i = \alpha + \beta x_i$$

Having said that, I don't see anything wrong with your notation. Beware, however, the likelihood can be a function of more than one parameter. In the above model, for example, the parameters you want to estimate are $\alpha$ and $\beta$.

You normally use $\theta_i$ to indicate that the probability of "success" depends on each observation, i.e. it is not constant across your sample. Logistic regression is a famous example of such practice. There you model each probability in terms of one or more explanatory variables:

$$\theta_i = \alpha + \beta x_i$$

Having said that, I don't see anything wrong with your notation. Beware, however, the likelihood can be a function of more than one parameter. In the above model, for example, the parameters you want to estimate are both $\alpha$ and $\beta$.