I have always taught my students that the $y$-intercept of a line is the $y$-coordinate of the point of intersection of a line with the $y$-axis, that is, for the line given by the equation $y=mx+y_0$, the $y$-intercept is $y_0$. I emphasize that that the $y$-intercept is the number $y_0$ and not the point $(0,y_0)$.
But I was quite surprised when I recently looked at the Wikipedia and Wolfram MathWorld entries for $y$-intercept because these define the intercept as a point and not as a number ("the point where a line crosses the y-axis" and "The point at which a curve or function crosses the y-axis").
Further investigation yielded inconsistencies: the Wikipedia entry for "Line (geometry)" states that in the equation $y=mx+b$, "$b$ is the y-intercept of the line"; the Wolfram MathWorld entry for "Line" states that "The line with $y$-intercept $b$ and slope $m$ is given by the slope-intercept form $y=mx+b$.
Edit made on February 21, 2021
According to the Dictionary of Analysis, Calculus, and Differential Equations (edited by Douglas N. Clark, published by CRC Press in 2000),
intercept The point(s) where a curve or graph of a function in $\mathbf R^n$ crosses one of the axes. For the graph of $y=f(x)$ in $\mathbf R^2$, the $y$-intercept is the point $(0,f(0))$ and the $x$-intercepts are the points $(p,f(p))$ such that $f(p)=0$.
Unfortunately, the book does not consistently use that definition.
slope-intercept equation of line An equation of the form $y=mx+b$, for a straight line in $\mathbf R^2$. Here $m$ is the slope of the line and $b$ is the $y$-intercept; that is, $y=b$, when $x=0$.
Thus, even though the book defines an intercept as a point, it uses the term to denote a number.
Is there a trusted source targeted at mathematics educators (from, say, a government agency, an educational institution, or an organization) that defines "intercept" and consistently uses that definition?