First of all, cooking times are always only guidelines. The rule is always, "Cook it until it's done". The USDA suggests a finished internal temperature of 165 F (74 C).
USDA recommends cooking whole duck or goose to a safe minimum internal temperature of 165 °F as measured using a food thermometer. Check the internal temperature in the innermost part of the thigh and wing and the thickest part of the breast. When cooking pieces, the breast, drumsticks, thighs, and wings should be cooked until they reach a safe minimum internal temperature of 165 °F.
So, the answer to your question is that you cook it until enough time has past that the goose is 165 F...
With the caveat that your target temperature may be lower than the USDA recommendations because we all love eating rare meat. From an episode of The Splendid Table, Hank Shaw mentions his recommended target temperatures for different parts of a goose:
HS: The first thing you need to know is the sweet spot of cooking a goose breast is somewhere around 135 to 145 degrees Fahrenheit internal temperature.
LRK: It's pink?
HS: It's pink, yes -- medium-rare to medium. The sweet spot for the legs is somewhere around 175 to 180 degrees Fahrenheit, give or take, they're pretty forgiving. That's a huge spread in temperature.
He achieves this temperature difference by removing the breasts half-way through cooking and returning the rest of the bird to the oven until the higher temperature is reached. Note, the temperature of the breasts is below the recommended temperature, so there is likely some amount of risk with eating it this cool.
The entire method is explained in the link and it's pretty fascinating. While he gives estimates of time (45 minutes for breasts plus 45 minutes more for the remainder) which are quite a bit lower than your two recipes despite being for a lower oven temperature (325 F) and a larger bird 12-16 lbs, it's possibly due to the target temperature being so low for the breasts and, once they're removed, the remainder of the bird would probably cook more quickly.
To address the methods you're looking at, I'm going to convert the recipes to be in the same units:
Ramsay - 450 F for 10 minutes followed by 9/15 min/lb at 375 F
Food Network - 450 F preheat then 20-25 min/lb at 350.
Now, let's look at your two recipes...
- Stuff the zested fruit and the herb sprigs inside the bird and set aside for at least 15 mins. Can be done up to a day ahead and kept refrigerated.
Rub inside cavity with lemon juice. Place apple, potato, orange and celery inside the body cavity. Truss the bird like a turkey.
I'm going to guess that the apple, potato, orange and celery combo may be more dense than seven lemons/limes. If it is, it's probably going to retard the cooking time a bit. This may explain some of your difference in cooking times, so you should keep that in mind. I don't know how you're planning to prepare your bird but the filling may change the cooking time.
Also, that big 450 F burst for 10 minutes is a big difference over simply raising the oven to that temp and immediately lowering it. When you open the oven, a lot of the air will leave, causing the air temp to drop a bit. If you immediately lower the temperature, it will drop to 350 F pretty quickly.
With Ramsay's method you're actively cooking the bird for 10 minutes at 450 F. This means that the oven will bring itself back up to 450 F and stay there for a few minutes before you drop the temperature to 375 F and it will take longer for the oven temperature to drop (assuming you have a well-insulated oven). The goose will start warming up more quickly with this method, which is reflected in the shorter cooking time.
Plus, when you're talking about 2 hours or more, a 25 F temperature difference does change the cooking time.
So, in the end, I think the methods are quite different but I think that (if you follow the actual method including the filling) both will give acceptable results... but that doesn't matter because you should at least find a recipe that tells you what temperature to aim for rather than what time to cook for and refer to those recipes' internal temperature recommendations and use your meat thermometer to know when you've reached them.