Not best practice but as someone who has done hiring before at other companies they often wont catch you were a previous employee if you dont specify it. To be fair if you quit without offering a 2 weeks during training the store probably marked you as non hirable on the basis of being unreliable.

I believe you do need to specify that you were a partner, but not certain on that. You may have been marked as rehireable but that all depends on who your SM was. Also, if you’re reapplying in the same district there might be someone that can vouch for you that worked with you albeit briefly!