George 13 b Suffolk, Eliza 11 b Portobello, Mary 9 b All Saints Southampton and Chartotte 3 b Greenland, Scotland [Greenland is near Inverness, nowhere near Edinburgh]
There are five other Greenlands in Scotland named on the 1:50,000 Ordnance Survey maps
Caithness
http://www.geograph.org.uk/gridref/ND2467Dunbarton
http://www.geograph.org.uk/gridref/NS4374Argyll
http://www.geograph.org.uk/gridref/NR7323Shetland
http://www.geograph.org.uk/gridref/HU2249Aberdeen
http://www.geograph.org.uk/gridref/NJ8118None of these can by any stretch of the imagination described as 'near' either Inverness or Edinburgh - they are all at least 100 miles from Inverness, and only the Dunbartonshire one is less than 100 miles from Edinburgh, so if you have found one near Inverness, and Charlotte says her birthplace is Greenland in or near Edinburgh, there must be at least two more.
The Midlothian Name Book of 1852/1853 lists a Greenland Mill in the parish of St Cuthbert's, which is in Edinburgh. It was a flour mill and it was 'about 8 chains ENE of Trinity Chapel'. 8 chains is about 150 metres.
This can't have been the Trinity Chapel which was roughly where Waverley Station is now. See
https://collections.ed.ac.uk/calendars/record/19466 and was demolished about 1848, because the same book says that Dean Bridge Cottage is 'about 4 chains NNW of Trinity Chapel'. This makes far more sense than Calton Hill, because the Dean Bridge (built 1831) crosses the Water of Leith, and the Water of Leith was lined with mills of all kinds - flour, barley, paper, snuff and other commodities.
Other mills listed with reference to 'Trinity Chapel' are West Mill, Mar's Mill, and Lindsay's Mill. So I took a look at some of the old town plans of Edinburgh and found
http://maps.nls.uk/view/74400935 which shows Trinity Chapel and some mills.
On the site of this Trinity Chapel is now Holy Trinity Church, built as an Episcopal church in 1837/1838. It was converted to an electricity sub-station, but I think is now back in ecclesiastical use by some minority denomination or other. See
http://www.geograph.org.uk/photo/2697521.
So I reckon that Charlotte's birthplace must have been beside the Water of Leith, downstream of the Dean Bridge and before you get to St Bernard's Well.
There's no record of her baptism in Scotland's People, but that isn't surprising. As an English family, they probably attended an Anglican church, and the Episcopal Church in Scotland records are not included in Scotland's People.
In fact, there's a good chance she was baptised in Holy Trinity Church. I have no idea whether its registers have survived, and if so where they might be, but if you are really keen to find out I suggest starting with the diocesan office of the Scottish Episcopal Church
https://www.scotland.anglican.org/. If you can find this, it should tell you the surname of Charlotte's mother.