You have it basically right. First an application is submitted and it gets an application number and a tentative filing date. Over the next few weeks it is checked by administrative people for conformance and completeness with respect to formal requirements. They are not looking at content other then that the minimum requirements are met. These include internal consistently between the drawings' figure numbering and the specifications list of drawings. If formalities are not met the applicant gets a notice to provide missing parts or correct problems with the submission (but they can't add new subject matter at all). A deadline to fix the application is given; if not met the application can go abandoned. In that case it is never published. The application is kept in confidence by the USPTO and there is no way for the public to tell it exists.
Publishing is typesetting whatever the applicant submitted into a form that looks like a patent, assigning it a publication number and making the document available to the public. It is then searchable on the USPTO site, on google patents and many other patent document databases. This generally happens automatically 18 months from the initial filing as long as the application has not gone abandoned before that. It looks like a patent but could almost be gibberish.
At some later point it is examined for novelty and obviousness. There is back and forth between the examiner and the applicant, all publically visible after the application is published. The back and forth generally includes objections, rejections, arguments and amendments. The ultimate result might be a rejection that the applicant chooses not to respond to causing an abandonment. Another result can be a notice of allowance. After paying money, the final content is typeset and given another number, a patent number. When finished, it is issued as a granted patent on some particular day and the document is publically released. The term "published" is not usually used in regard to the issuing of the patent.
Issued and granted are almost synonyms.
This is a somewhat simplified explanation.