Q1: Ringel defined, for any given quasi-hereditary algebra $\Lambda$, a particular tilting-cotilting module called the characteristic tilting module. This characteristic tilting module has both a standard and a costandard filtration. Due to a slight misunderstanding, Donkin used the name 'tilting module' for the summands of this particular module. These two concepts are almost never the same, since for any given algebra there are usually many different (generalized) tilting modules in the original sense. For instance is $\Lambda$ itself always a tilting module.
Q3: There is a concept called Wakamatsu tilting module (see Handbook of Tilting Theory, page 207) where you leave out condition 1), keep condition 2), and in condition 3) replace the sequence with a possibly infinite coresolution.