Settling 1 square to the south
I like the idea of settling 1 square to the south. Consider that if the Forest wasn't there, you'd almost certainly move there on the first turn in favour of settling at the starting location. So, there's just an extra bit of pain involved here, of having to use up one extra turn, to make you hesitate.
Settling in place means that you won't get to use the Grassland Hills square, on which we start, for extra production. That square could be very useful with so many Grassland squares around the starting location. Will that extra turn really make up for a lack of Hammers in your capital?
However, settling 1 S:
a) gives you that Grassland Hills square
b) keeps you next to the Lake for the Fresh Water bonus
c) keeps you on the Coast for sea-based buildings and for building ships
d) better secures the Gems without needing a second city. At this difficulty level, getting 2 or 3 cities total is about all that I can hope for early on, so I don't want to have to commit a city location just to grab that Gem Resource when I can get it for free within my capital's fat cross
e) moves you closer to the Ivory squares so that they are harder for an AI city to culturally steal
f) allows you to keep all of the visible Resources within your city's fat cross
g) gives you a better chance of choosing when the AI to the east meet the AI to the west (assuming that there is at least one AI to either side of our starting location). I'd rather block the AI off and then choose who to let by with Open Border agreements--it'll be a while after Writing before you get Alphabet anyway, so I don't worry about the AI not getting a chance to meet each other, should I chose to let them do so
h) trades 2 Grassland squares, 2 Coastal squares, and 1 Ocean square for 5 Grassland-based squares, including 2 squares with a Resource on them. Yes, those 5 squares all appear to have Jungle on them, but Jungle can be cleared. An Ocean square won't be of much use unless Oil appears there; if I am lucky enough to get that far in tech without losing my capital, I'll be happy to have the Oil appear just outside of my fat cross while still being in my Civ's cultural borders
Settling 1 square to the west
Yes, settling 1 W on top of a 3 Food Resource is also a viable option.
However, there is an additional consideration with losing out on the abilities of building sea-based buildings and building ships. You will miss out on building a Lighthouse for the Lake. I'd be willing to have that Lake as a 3 Food, 2 Commerce square, so that maybe I can build a city close by to the west. The capital could use the Lake square while the city to the west shares the Pig and maybe some of the other Food Resource squares within our capital's fat cross.
I'm also afraid that an AI would take advantage of our move to plop down a city one square to the east of the east-most Ivory. With some cultural pressure, they could feasibly steal all of the Ivory squares from us.
Even with 1 Ivory square, an AI is likely to get a larger army of War Elephants than us, and likely faster than we can even research Construction, so I'd be very hesitant in giving that obvious opening to an AI.
Sure, you could put YOUR second city one square to the east of the east-most Ivory, but are you certain to settle that location before an AI does? That proposition sounds pretty risky to me, at this difficulty level.
Other random comments
Settling on the Ivory to the SE won't gain you much, because the Ivory is on a Grassland square. If it were on a Plains square, you could net yourself an extra Hammer in your city's centre, but fortunately, we aren't faced with that difficult choice.
The last post to this thread was "By Jove," which I found to be pretty funny, since there are several sayings that start with those words. One which springs to mind is: "By Jove, guv'nor, I believe he's had away with your wallet!"
If I had a choice between trading for Mathematics or Iron Working from Alphabet, I'd take Mathematics. Obviously, if you can get both, do so. We know for certain that we have Ivory and Construction is only two techs away from Mathematics (try to trade for Masonry so that it's only one tech away!). While Praetorians are beauties, getting Iron is likely going to require you to settle another city after you've learned Iron Working. Yes, we might have Iron on our starting Grassland Hills square, but people did ask for a non-doctored Deity level game, so I'm not counting on it.