Does the inclusion of a model offset convert predictor variables from counts to rates?

Does the inclusion of a model offset in Poisson or logistic regression convert predictor variables from counts to rates? Or does it only convert response variables from counts to rates?

I understand that offset variables are used in Poisson or logistic regression to account for differences in sampling intensity/exposure at each observation. The inclusion of an offset variable in Poisson or logistic regression allows you to model rates instead of counts. I understand the conversion from count to rate is done by including the units of observation as a predictor variable with a fixed coefficient/slope of 1.

A hypothetical situation: To model the prevalence of virus_1 in a population of animals, I would build a model like below, where my response variable is the count of animals positive to virus_1 at each observation, and my offset, which converts this count to a rate/prevalence, would be log(total_number_animal_sampled). However, how would I include the prevalence of a second virus as a predictor variable in this model? If I was to include count_virus_2_positive as a predictor variable would this count also be converted to a rate/prevalence because of the inclusion of the model offset?

Note that the below model also includes predictor variables that are not influenced by sampling intensity/exposure at each observation, for example the abundance of animals at site at each observation.

glmmTMB(count_virus_1_positive ~ offset(log(number_animals_sampled)) + count_virus_2_positive + count_virus_3_positive + site_abundance, data = data, family = binomial)


The variables (response and predictors) are what they are, the use of an offset does not change the variables. What it changes is the interpretation of the results.
So in your example, the inclusion of predictor count_virus_2_positive will be as a count, not as a rate.