Difference between physical, proper, and co-moving distances

Eq. (1.14) and (1.15) of David Tong's lectures (p.g. 12) writes:

Consider a galaxy which, in co-moving coordinates, traces a trajectory $$\vec{x}(t)$$. Then, in physical coordinates, the position is $$\overrightarrow{x}_{\small{\text{phys}}}(t)=a(t)\overrightarrow{x}(t)\tag{1.14}$$ The physical velocity is then $$\overrightarrow{v}_{\small{\text{phys}}}(t)=\frac{d\overrightarrow{x}_{\small{\text{phys}}}}{dt}=H\overrightarrow{x}_{\small{\text{phys}}}+\overrightarrow{v}_{\small{\text{pec}}}\tag{1.15}$$ where $$a(t)$$ is the scale factor and $$H=\dot{a}/a$$.

My question is this:

Assuming that $$\overrightarrow{v}_{\small{\text{pec}}}$$ is constant, can I write $$\overrightarrow{x}_{\small{\text{phys}}}(t)=a(t)(\overrightarrow{v}_{\small{\text{pec}}}t+C)$$ where $$C$$ is a fixed distance. If so, would that make $$C$$ the co-moving distance? Thus, $$\overrightarrow{x}_{\small{\text{phys}}}(t)=a(t)\overrightarrow{v}_{\small{\text{pec}}}t+\overrightarrow{x}_{\small{P}}(t),$$ where $$\overrightarrow{x}_{\small{P}}(t)=a(t)C$$ is the proper distance?

Consider a stationary point in comoving coordinates (i.e. a comoving observer), call it $$\vec{x}_0$$. The trajectory in proper coordinates is then $$\vec{x}_{phys}(t) =a(t) \vec{x}_0$$, which is just your first equation with zero peculiar velocity. So it makes sense to interpret your $$C$$ as a (fixed) comoving coordinate. The $$\vec{v}_{pec}t$$ part is problematic, though, because any physical distance traversed due to the peculiar velocity subsequently expands, but only after having been traversed - you can't avoid an integral in this equation.