That extended explanation is more accurate, yes. I'd call your points 1 and 2 both training under the definition "anything that adjusts model weights is training". There are multiple stages and types of training. Right now AFAIK most (all) architectures then fix the weights and you have non-weight-affecting steps like the system prompt, context, etc.
You're right that the weights can enable the model to memorize training data.
You're right that the weights can enable the model to memorize training data.