Yes it is true. It’s a list of spent transactions, not a list of dbcs that clients are able to spend.
Transactions that are written to the spentbook are now spent. The stamp of approval that comes after that from the mint only adds authority, but does not change it from being spent. Once the client writes to the spentbook there is no second chance to spend it differently, so it’s spent.
The literature for dbcs uses spendbook, I personally feel it’s a spentbook, and the difference is important for building an intuitive understanding of what this thing is.
Spend is a verb. It’s implying that the book lists something that’s ‘in motion’ or ‘happening’. This isn’t the case here.
Spent is an adjective. It’s describing the state of the data. It’s in the past, spent, not able to be changed, which is an accurate description.
To use a bitcoin analogy, spendbook would be utxo (ready to spend, able to spend etc), which the dbc record is not doing. The spentbook is any transaction that’s completed, ie the bitcoin blockchain not including utxo. It’s an important distinction since it affects how we can use the data, eg can it be used for an audit or not? Can it be used for linking history or not?