Laura,
The answer to your question is - it depends. Each weight uses different set of variables, but the wave 2 weights will combine correction based on roughly between 50 to 100 variables of different kinds. I hope the details below will give you a better understanding of what this may mean for your estimation.
The weights in wave 2 consist of a number of corrections, all of which are important. The first part is design weight, which reflects the design and is not calculated based on any of the variables in the dataset. The second part corrects for household nonresponse at wave 1. This is a large part of the weight value, and the variables used for modelling this part are not released in the dataset - most of these variables are neighbourhood characteristics from Census and other administrative datasets. The next part, the correction for the individual within household nonresponse, is modelled using information from household grid and household questionnaire at wave 1. My understanding is that you are mainly worried about information from an individual questionnaire used for weighting. If you are an advanced user, you may want to model nonresponse yourself (possibly as part of your model). In this situation I would suggest to use wave 1 cross-sectional relevant weight, and correct for attrition between wave 1 and wave 2 as part of your model.
Just a warning note: some users think that one may control for nonresponse (or attrition) by putting control variables related to nonresponse into the model. This isn't true. Such approach corrects only for nonresponse in the intercept (mean or proportion estimate), and does not correct for the difference between respondents and nonrespondents in other estimates (for example regression coefficients), neither it reflects the correct standard errors of such estimates.
In my opinion you should not worry if the weights use the same variables as are present in your model. If the relationships between variables are influenced by weighting, they are changed in the direction closer to the relationship in the population - i.e. to reflect that nonrespondents tend to be with a different relationship between these variables.
I hope this is helpful,
Olena