Does my 10-year ban end on the 1 January 2019, or on 11 September 2019?
No. If you have a ten-year ban, it ends at some other time, unless you were deported on the day you were caught.
There is not enough information to answer your question, but we can offer some possibilities.
If you were deported or removed from the UK, you have a 10-year ban from the date of the deportation or removal, as noted in Dirty-flow's answer.
If you received a prison sentence that was at least 12 months long, but shorter than 4 years, you have a 10-year ban from the end of the sentence. So, for example, if you were sentenced to a 47-month sentence beginning in 2010, and left the UK voluntarily after the end of the sentence, your ban ends in late 2023.
In both of these cases, the beginning of the ban will have been sometime after September 11, 2009, so in neither case will it be January 1, 2019 nor September 11, 2019.
If neither of those cases applies, you have a ban with a different duration. For example, if your sentence was 4 years or longer, you have a permanent ban. If your sentence was less than 12 months, you have a 5-year ban. If you were removed "as a condition of a caution [under] the Criminal Justice Act 2003," you also have a 5-year ban.
The details are in Appendix V of the visitor rules.