You are misusing the rule based labeling.
The formula in the "rule" setting is used to select witch object are labelled, not how they will be labelled. So your two rule should be :
Description : Timestamp value is 1 : bigger and black font - Rule :
"Timestamp" = 1
Description : Timestamp value is 0 : smaller and grey font - Rule :
"Timestamp" = 0
(It could be advisable to set a third "ELSE" rule if you can't guarantee that the timestamp will always be filled with either 0 or 1)
Then you set how each rule are labeled the exact same way you would while using single labels . Each rule are totally independent on how they are labelled and can use any setting regardless on setting on other rules. It's the same as if you where labeling two different layer.
If you want to use data defined override you should select single label, set all labeling option the usual way except for the font size and font color where you should select the edit option by clicking on the symbol next right to the font and symbol selector and use an expression to set the relevant setting (for font size you may use the
CASE WHEN "Timestamp" = 1 THEN 10 ELSE 5 END expression suggested by @Erik)
Note that when using data defined override you need to input an expression for each setting you want to individualise while rule labeling require that you create a rule for each kind of label you need.
So as a rule of thumb : for creating a few distinct label type (as in your case two distinct label) it's better to use rule labeling. While if you want to get lot of similar label (like a label colored accordingly to a field value) it's better to use data defined override. Of course for overly complex labeling you may mix the two approach....