How to describe the process of using existing weights for training a deep learning model?
Yes it is definitely doable, and further more, its is recommended in many scenarios. For instance, if you wanted to train some generic image classifier it would be naive idea to start from scratch and, for instance, retrain ImageNets' dataset for 2-3 weeks simply trying to retrieve optimal weights that have already been previously computed. 

if you simply take pre-trained weights and use them on a different domain, then that would be referred to as transfer learning. However, since you want to retrain the existing weights with additional training data then, yes, you would be "fine-tuning" the model. When you do this however, there are considerations that need to be taken. Do you want to fine tune every layer? Or only deeper layers that detect very specific features of the dataset? Considerations like these should be taken into account. 

Also check out http://cs231n.github.io/transfer-learning/