Just to add further clarification in case it helps - in last week's paper on aspirin and prostate cancer death - in the Cox proportional hazard model, I think they treated aspirin as a time-varying exposure, with simple updating (until the last cycle, when it was lagged). this means if they say they are not using at baseline, then for the next follow-up cycle they are a "non-user"; then at the next survey, they say they started aspirin, so their exposure for the next follow-up cycle is updated to "user".
All those person-time diagrams we were putting on the board last week were examples of time varying exposure.
A time-varying covariate is applying these sample principles to update information about other variables in the model that could change over time. For example, maybe BMI or age could be updated, if one was very concerned about adjusting for confounding by those factors.
best,
JMC