To reinforce further my comment, as well as what Marty and Geoff say, the notion of "defect" of a $p$-block in a group algebra over a sufficiently large field of characteristic $p>0$ was developed by Brauer in a character-theoretic setting. This is presented in the last part of the classic 1962 book by Curtis and Reiner,
*Representations of Finite Groups and Associative Algebras*. This got developed in other styles over the years in work of Green, Alperin, etc., so for instance Alperin's small book *Local Representation Theory* takes mainly a characteristic $p$ viewpoint without emphasis on ordinary characters. But the notion of height of a character in a $p$-block goes back mainly to Brauer's work and relies on the study of ordinary characters.

Here is a small example which illustrates why the choice of a height 0 character will typically not be canonical (except for blocks of defect 0). Consider the smallest nonabelian simple group, of order 60, realized for example as $\mathrm{PSL}_2(\mathbb{F}_5)$. Here the ordinary irredudible characters have degrees $1, 3, 3, 4, 5$, with the squares of the degrees of course adding up to 60. As in other simple groups of Lie type, there are only two $5$-blocks here. The principal $5$-block has defect 1 (the highest power of 5 in the group order) and involves the first four characters; since $p=5$ divides none of their degrees, all have height 0. But the (Steinberg) character of degreee 5 lies by itself in a $5$-block of defect 0 and has height 0.