I DID try "AT+UART_DEF" and "AT+UART_DEF", my module was returning "ERROR" - which was part of my main confusion.
I became confused on the part of updating the firmware. Perhaps I had been spoiled in the past, just by going to a git-hub and finding the most recent one (in ENGLISH).
Is not obviously labeled "at command set". I must sort through various online tutorials which all use different examples which have different ways to accomplish the same basic thing, sorting "esspressif" hub scrambled with CHINEESE:
"NONOS_SDK"
"IOT_PLATFORM"
AT_v0.50_on_esp_iot_sdk_v1.4.0_150918"
"esp8266_nonos_sdk_v1.5.4_16_05_20"
"ESP8266_AT_V00180902_02_baudrate watchdog added"
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In the directory of "the most recent" firmware, there are TWO sub-directories "AT" and "AT_SDIO". At this point is a nice gamble in time for me to figure out which one does not work. The were no instructions or obvious information for me to pick, possibly due to my patience running out reading so many different readme's at this point I was really frustrated. I had to load previous versions based on the information in the year+ old tutorials, and failed with those attempts.
In each "README" warns I must load something else PRIOR to loading THAT firmware, and then go on a scavenger hunt for a bunch of other files to study and load - which is very time consuming (for me).
The nano uses software serial for 2nd UART, that I must use in order to send my debug messages back, with a great pleasure of swapping wires on a breadboard back and fourth, unplugging serial cables and such just to update my .ino project over and over again.
As of last night, I believe I loaded the most recent firmware, tested my module was able to connect to my router to obtain an IP address. I can't remember if I tested changing the baud rate.
So let me ask this, if I can't change the baud rate - I must not have loaded the at-command set? I was getting reply "OK" from "AT" and returned firmware version "AT+GMR", so I thought THAT WAS the at command set? I think there are different flavors in the same version of firmware for different at-command sets.