I agree that it depends on the therapist, their boundaries, their availability, and whatever agreement exists between therapist and client.

It seems, Scarlet, that you and L have the sort of agreement where L generally provides crisis support. However, it sounds like L has had inconsistent and unclear boundaries lately. The stuff about only email if contacted by email and only call if contacted by call sounds rather rigid. And also like she's been inconsistent with that.

If she doesn't feel that she's able to be there as much as you might want or need, then I feel she needs to let you know that up front. Even if it's a case of her saying "the majority of the time, I will be available to call or email, but sometimes I won't be available, and here's what you should do if that's the case." Because no one can be available at all times.

Also, in the case of her not being able to call/email until 7 that one day, is it possible she had other clients she was meeting with all day? I know some therapists are willing to reschedule (or be late to) another client to deal with one in crisis, but others are not. And I know it can be difficult to think that a therapist is prioritizing other clients. But I think of the time that ex-MC took a cell phone call from another client while H and I were in session with him. He was only on the phone for a couple minutes, but it was distressing to me that he opted to talk to another client while we were sitting there in the room with him (he did take it in the courtyard). I think it could help, Scarlet, for you to think about how you might feel if L was late to a session with you (or even canceled) because she was helping another client.

That being said, she should NOT have told you that emailing you made her 5 minutes late meeting with another client. That's on L (the decision to email you and push back her other client). She could have chosen not to respond to you right then. In choosing to respond to you, she should not have told you that she was late for another client because of you. It was her decision, and you're not responsible for that other client--L is. In the same way that you're not responsible for holding boundaries--L is.

I get the sense that L is struggling to uphold what she offered you in terms of crisis support, phone calls, emails, etc. She could be overextended or burned out right now (not your responsibility). Whatever the reason, it sounds like she can't be there in the way you want/need right now. So I think that seeking out other support, as Electric Manatee mentioned, could be a good option (I've done that when I have had ruptures/conflicts with my T, and it has helped). Including reaching out to T.

But you also may need to have a discussion with L, assuming you opt to work things out, about what realistic expectations are, when she feels she can call or email, with what frequency (and maybe what times of the day/what days), whether you need to contact her in a certain way to get a call, whether you need to explicitly ask for a call if you want one (vs. saying you're unsafe), etc. And you may need to adjust your expectations and wants for what she is willing to offer.*

I hope you're able to work it out.

*My T used to have a fairly open schedule (he opted to have a lighter caseload, like 30 sessions per week), such that I could ask for an earlier or extra session the next day, and he'd nearly always have an opening (or might add a session). With the pandemic, and a high need for mental health care in our area, he's taken on a greater caseload of clients than he'd prefer to have (like 40 sessions per week). So now he's often not available for an earlier or extra session. Which has been difficult for me to adapt to. But we've discussed it, and I've come to realize that it's just how things are right now, and I can't assume a session will be available if I want one.