Do the prediction errors mentioned in the slides refer to training or test errors?
Systematic predictions error
It could refer to both training and testing errors. Indeed, an underfitting model will be stable because it is not flexible. Thus, it will make systematic errors in both training and testing sets.
predictions errors without obvious structure
It refers here to the testing errors. Indeed, an overfitting model will “perfectly” fit the training data and thus fit noisy samples (and very few errors will be committed). On the testing set, the model will commit errors. However, changing slightly the training dataset will change widely the model learned. And thus the errors on the subsequent testing set will always be different, without a systematic (obvious) pattern .
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The last word you meant underfits?
The last word you meant underfits
Indeed thanks a lot, I edited @glemaitre58’s post and reword it a bit in order to remove the confusion!
Indeed. Thanks for editing my post 
Thanks for the explanation. One normally associates “prediction” error with test set. This association also holds in this case.
It’s just that for underfitting models, even the train errors are high and have similar attributes as the test errors.