What is the definition of a stringent/relaxed plasmid?
In short, yes, the definitions are still correct:

The number of copies of a plasmid in the cell is determined by the mechanism of its replication: whether it is synchronized with the replication of the bacterial chromosome or is independent of it.

In the first case, the initiation of replication is performed by the same mechanisms of replication of the bacterial chromosome. This means that the origins of replication, as well as all proteins involved in the initiation of replication, are shared between the plasmid and the bacterial chromosome. For plasmids, which replication is synchronized with the replication of the bacterial chromosome are under stringent control of replication. The stringent control of replication is characteristics of plasmids with low copy number. These plasmids contain DNA regions called par-locuses (from partition) that control the correct division of the plasmids in the two daughter cells.

Plasmids with high copy number are under relaxed control of replication. These plasmids are replicated independently of the replication of the bacterial chromosome. They have individual mechanisms for initiation of replication. The plasmids themselves encode the initiation factors for replication, which are mainly RNA molecules. Once the plasmid replication is initiated, it continues with the same machinery used by the bacterial chromosome. 

Reference (Textbook): Bacterial Genetics, Margarita Pesheva (2000)