Linear Regression a.k.a. Ordinary Least Squares assumes only Y has noise, and X is correct.
Your "visual inspection" assumes both X and Y have noise. That's called Total Least Squares.
Yep, to demonstrate, tilt it (swap x and y) and do it again. Maybe this is what TLS does?
There is an illustration in https://en.wikipedia.org/wiki/Total_least_squares