Don K:
I travel for business and fly pretty much weekly - going on 11 years now. I'm not sure what that site is or where whoever wrote that article got that information - perhaps it is just old...but I can assure you that you can indeed bring those items on board in your carry on luggage. There is no restriction on type of liquid (except the obvious ones like explosives and such) so you can bring sunscreen, body lotion, shaving gel, hair spray, toothpaste, etc.
However there are limits on how much you can bring.
You are allowed to have liquids in bottles or tubes up to 3.4 oz in size. Note that the bottle size must be 3.4 oz or smaller - having a larger bottle or tube which is half empty doesn't count (they will take it from you - I learned the hard way and lost my expensive bottle of hand lotion that way back when all this started).
All of your bottles/tubes then have to fit into one quart-sized zip-top baggie. I have found that different airports have different definitions of "fit" - some expect that each bottle lays flat and neatly side by side - others allow bags which are stuffed full and barely zip closed. Most airports have clear bags that you can use (you find them at the beginning of the security line usually) instead of your own zip-top bag. I've never used theirs, but it seems that they are a bit larger than the standard quart-size bag.
I never put my mascara in the baggie, and I've never had a problem. But ladies, remember to check your purse for any "offending" items prior to going thru security - I have gotten stopped because I forgot about my hand gel or eye drops in the back pocket of my purse. (though truthfully I've gotten thru dozens of times only to realize later that I did have one of those items in my purse or computer bag and wasn't stopped.)
In addition to the one baggie, you are allowed to bring thru liquid medications, baby formula, eye drops, nasal sprays, etc. in a separate clear bag. I always have a second bag for my contact solution and eye drops.
If you are unsure about sizes or quantity of your liquids, then check the bag. If you go thru security and they find something that you are not allowed to carry thru, they will give you the option of surrendering the item or going back and checking your bag.