The controller was incorrectly telling other aircraft that the OP turned base on their own. Then the controller told the OP to listen up next time. If a controller has a problem with what a pilot does you say "possible pilot deviation, advise you call me on xxxx after landing." It's either a PD or it isn't.
The OP received a "continue" on downwind with no other restriction. Two problems with this. First, "extend downwind" is the only phraseology given in the 7110.65 to adjust for sequencing and spacing for the OP's situation. Generally controllers throw in a "I'll call your base" with that. "Continue" is used when the controller is temporarily withholding the clearance for some reason (3-10-5 f.) or they aren't sure about the sequence and just want the aircraft to keep coming inbound (3-10-1). The pilot is expected to continue on a normal pattern and either report something or expect their clearance at a certain point. Second, the controller used "continue" by itself without further instructions. Years ago in ATC that was completely acceptable but now the FAA wants instructions after the "continue" to reduce confusion. That is why last year they added the note in 3-10 saying that they should provide additional information.
To the OP, you did exactly what I would have. I never even would have questioned the clarity of a "continue." When traffic is that busy, you don't need to be asking "OK, are you saying to continue on downwind or are you saying to continue inbound my approach?" Even the "Continue on downwind, I'll call your base" that she used with the next aircraft was improper. I'm not saying she is a horrible controller either. By the sound of it, she's been doing ATC for more than a day. I'm just saying she needs to tighten up her phraseology so it's more in line with the guidance in the 7110.65.