What is the name of the form you’re supposed to get after you complete AIT (not the certificate)?

I gather that you're talking about the DA Form 1059 that is generated by the school when you attend a course. The DA Form 1059 has all the applicable information regarding your time there, how well you did in a variety of areas, etc. It's the academic version of an evaluation report.

You'll see it appear in your records probably 3-4 months after you finish the course. The schoolhouse has 90 days after your graduation date to submit it and then it will go through processing.

However, your completion status should be posted by the school within 2 working days after graduation and your Military Education Level/Status should be updated in your records within two weeks of graduation. If it hasn't happened within two weeks, have your personnel folks get in touch with HRC's Military Schools Branch.

There will be orders uploaded into you Iperms from your training unit stating that you have completed AIT, if you are an IET soldier. If you are a reclass soldier there will be a course eval uploaded into your Iperms showing completion and scores if applicable. You could pull them from Iperms if you want them printed. However, your incoming unit will have access to all these items. You can log into Iperms at any time and review your paperwork, I actually encourage you to do this often.