Yes that is correct. Classifier is really telling the software what you would like it to do with the packet. But it does not change any fields in the packet header. The marking is applied only with the rewrite rule, so without a matching re-write rule the default will be applied based on th einterface and the device. SRX and Switch have different default and depending on interface type (trunk/access).
The classifier is like a gateman who peeks in the door and say SAM is a VIP, I am giving him a card that you should stamp him with #9 (forwarding class) and send him to booth 9 (queue) where he can get MOET and whatever else(special treatment). But only the Manager (rewrite-rule) can do that. Without the Manager and the stamp(marker for the forwarding class mentioned in the rewrite rule ), he only has the card telling the Manager what to do. But no Manager is there to do it. So he just defaults to the bar. First come first served
i.o.w the card (classifier) means zilch without the Manager(re-write rule).