Is Sda a protein, or is it a protein domain of DnaA?
Short answer

Wikipedia has unnecessarily made an already confusing situation much worse. Sda is its own protein, it is regulated by DnaA, and it prevents sporulation.

Full answer

I don't think your misunderstanding is based on the meaning of 'domain' - the definition you give sounds quite reasonable - but rather the use of the term domain with respect to Sda and the naming of Sda itself, which I agree is quite confusing and made much worse by Wikipedia in this case.

Sda is a protein that inhibits KinA activity (Burkholder et al., 2001) and is a completely separate protein from DnaA. I don't see where you jumped from your notes to the idea that Sda is a domain of DnaA except for from Wikipedia: I don't know why Wikipedia thinks Sda is a protein domain, it is a protein. It is a very small protein, so perhaps it would be appropriate to think of it all as one domain, but there is not a single result for "Sda protein domain" on Google Scholar - no one else is using this terminology. A full Google search only returns a few (~150) results, most of which are direct copies/translations from Wikipedia, or Wikipedia is a direct copy of them. Wikipedia can be useful, but if it starts to cause you confusion, don't assume it is correct.

The phrasing "KinA is bound and destabilised by Sda, a DnaA target" means that Sda binds KinA and destabilizes it, and notes that Sda is a target of DnaA; since DnaA is a transcription factor, you can infer that this means that DnaA controls expression of Sda in some way. 

In biology a checkpoint just refers to a place that a process can be halted/arrested, often in reference to the cell cycle. Sda is considered a checkpoint because when Sda is expressed, it prevents cells from sporulating by indirectly preventing activation of Spo0A (Burkholder et al., 2001).

Another source of confusion is that the name "suppressor of dnaA" on Wikipedia is WRONG, but further, molecular biologists sometimes have an impenetrable way of naming things, owing to the complexity of control of gene expression and the circuitous way that new proteins are identified and understood.

The correct name for sda is "suppressor of dnaA1". dnaA1 is NOT DnaA - dnaA1 is a MUTANT ALLELE of DnaA that results in cells that are completely unable to sporulate. But! sda is also NOT Sda: sda is a MUTANT ALLELE that produces non-functional Sda (the protein).

I'll quote from the title of the table in Burkholder et al:


  sda Mutations Suppress the Sporulation Defect of dnaA1 Mutants


Or, stated again, in dnaA1 mutants, there is a defect in DnaA which prevents sporulation. The researchers searched for mutations that would reverse this effect. They found one, and named it sda, because it reverses (suppresses) the effect of the mutation they were studying. The dnaA1 mutant's effects on sporulation seem to be caused by an increased expression of Sda, which is why the sda mutant reverses the effect.

References



Burkholder, W. F., Kurtser, I., &amp; Grossman, A. D. (2001). Replication initiation proteins regulate a developmental checkpoint in Bacillus subtilis. Cell, 104(2), 269-279.