I worked as an intern ASM and then an apprentice ASM. I essentially did everything a non-equity ASM would do, minus decent pay. (Props tracking, line notes, run shifts, supervise crews, etc) I couldn't get a call back on my resume until I dropped the "apprentice" part from my resume under the suggestion of my SM. I did that and got two calls the next week. People don't want to hire you if they think your last job was training, but fewer and fewer places are hiring non-equity ASM's in favor of the cheaper interns. I think it's more important to properly identify your responsibilities than properly name your title.