Yes you are overthinking it. You’re late. Your manager coached you about it. Start getting there on time. Even if you have to arrive super early and wait. It’s pretty basic stuff.

Really the only thing you have going for you in this post is that your SM seems not to mind a few minutes of tardiness, but otherwise, this is all on you. Your post makes it sound like you are pretty much brand new to the company, so here's some more information about management's role.
1. ASM's are not only in charge of management - they are there to learn to be an SM, so your ASM giving you shit for being late is totally within the realm of their responsibilities. I will say that I'm not sure why you would have to tell them if you were coming in late for a particular shift because that should lie with the shift (who is well within their rights to tell the ASM).
2. Even if your shift doesn't care that you were late (and if you work in a high volume store like I did, trust me, they DID care that you were late even if they didn't say it), company policy is company policy.

Most of the staff at my old store took buses to get to work in a busy city environment - they still could not be late. It is the same in the restaurant world - early is on time, on time is late, and late is unacceptable. Other than truly unimaginable circumstances, this always holds.

Speak to your SM then speak to your DM. If all else fails call PCC.

You're at fault here. Like... 100%.

Take responsibility. You clocked in 11 minutes late, THEN got ready to actually work. I'm thinking by the time you got on the floor it was 15 minutes late. How is that okay??? And now you want to transfer? Who in their right mind would take you on when you're not reliable.

Stop. Being. Late.

The how doesn't matter, just do it.

More of a side bar, but your drains should really be cleaned every morning. Trust me, they get really gross really quickly when they aren't (especially cold bar with all the berries). It could be that your ASM is trying to change a routine to bring it up to standard.

OP said their manager was ok with the lateness until they get situated. So hence why I said speak to their manager first. I just listed the order in the chain of command. It’s not about escalating their situation. Some partners don’t know they can reach out to higher ups if they have a issue anything.

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It isn’t fair to the people you’re relieving if you’re late.

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I had a similar situation where i couldnt afford taking the bus so i had my brother give me rides to work. He’s so lazy sometimes and his two jobs are a lot less time constricted that he didn’t get what it meant to be late to my shift. i started telling him that my shift begins at 1 when it was actually at 2:30 :) that way i arrived with enough time to drink my tea and read a lil before my shift. My point is that this is temporary (the timing thing) so just keep it cool. And if your ASM keeps going up your a$$ bout stuff like that just look at her in the eye and give her a couching conversation about being uninspiring and more or less hindering your everyday tasks. If she babbles just take it up to your SM calmly. Keep it professional and keep mentioning stuff from the starbucks handbook. You can always let off steam with friends but keep your head up boo ;)