I had a tube of Phil for years until it ran out. Now I just use marine grease I've had for years.
For rubber applications I have tubes of Silglyde, Molykote 55 o ring grease, and Honda Shin Etsu. Dielectric grease for electrical stuff.

Geezzzzzz... How many different lubes have I used over the years? And how many different things do I use grease on? Cars, Boats, Bicycles, Appliances, Fence Posts, even some parts of my fire arms... Ha! Now days for me there is only one grease.

Marine Grease

Do note that I do not know what the best grease would be for a Carbon Bike...

There is no difference. The "lithium" in lithium grease is actually from the reaction of lithium hydroxide or lithium carbonate with a triglyceride. Triglycerides are an ester of three fatty acids and glycerol and are obtained from vegetable fats or animal fats. The lithium base splits the ester into glycerin and a salt of the fatty acid chain. Your grandparents (or parents depending on how old you are) may have done this kind of chemistry to make soap. They may have used actual lye (sodium hydroxide) or just ashes from a wood fire. The result is a fairly harsh, hard soap.

Lithium grease takes this "soap" or surfactant and uses it to dissolve up oil in the solid to make grease. The mixture is likely 90% oil and the rest the surfactant and/or other stuff. The resultant mixture will remain stable for a very long time. Over time and exposure to the elements, however, the oil can start to bleed out. The surfactant is left behind. None lithium grease can be a mixture of surfactants or it can contain clay or both. When people open up old bearings and find "peanut butter" or "grape jelly" inside, what they are seeing is the binder (soap or clay) left behind. The oil didn't "harden", it flowed away over time.

Modern polyurethane greases use polyurethane as the binder. It still uses oil but the oil seems to hold in the polyurethane better than soap based greases to. It's more stable for longer.

How long it takes for the oil to separate from the surfactant depends on how well the grease was emulsified in production. Cheap greases will separate faster than better greases. My experience is that good quality grease like Pennzoil #705 and Campagnolo grease can go years without separating, while bargain auto parts discount store grease can start to separate in a few months.

N.B. this separation issue is a reason why many old-school cup-and-cone hubs have an oil port where you can add more oil to replace that which leaked out. The act of riding will re-emulsify the oil with the surfactant and restore the grease.

Perhaps but most people will..and did…give the advice that “grease is grease”. That was also likely the reason that the suggested interval for greasing bearings was about 6 months or twice a year. Soap based greases could, in theory, wash out because the soap is a soap and can be slightly water soluble.

Modern polyurethane greases don’t have that problem because they don’t really separate.

That’s really, really, really, really old school. I seldom see a hub with the ability to add oil to the hub and I see a lot of what could be considered to be antique wheels.

Dang. You're making me feel really old!

Yea, back in the day, we used to rebuild our Campy Record hubs, headset, pedals etc. every winter. It gave me something to do, but the white grease always looked white and clean at the end of the season.

Now, I mainly use the white stuff in a syringe when I work of cheap or kids bikes. Back off the cones a little, squirt some stuff in and readjust.

​​​​​​​As soon as I use my tub up, I will buy some "proper grease." At this rate, that should be around 2075.

I use a grease gun with a needle tip. It's long enough to reach the far side of the bottom bracket, if I don't want to be bothered removing the fixed cup: