The older birth and death registers have been scanned and you generally get an image of what has been submitted to the GRO from the local registrars.
If the film is badly faded/illegible you will get a handwritten copy but they are few and far between. You now sometimes get a typed copy if the original is in a typed format. These tend to be much more recent though.
I've yet to come across an error of the nature you are suggesting ie the register office or GRO/Somerset House supplying incorrectly transcribed information on birth and death certificates. But I'm sure there are people here who will tell us otherwise.
The errors generally come in the compilation of the indexes.
Of course, once you have the certificate from the GRO there is nothing to stop you contacting Liverpool reg office and ask them to check their entry. It may be easier to write or email and enclose a photocopy or attach a scan as they may think you are just trying to get birth information from them without paying.
