Question about the the velocity and acceleration in tensor notation

Definition of directional covariant derivative
The directional covariant derivative of a vector along a curve in a manifold is defined as
$D/dsigma P^mu$
where:
$D/d$ directional covariant derivative
$gamma (sigma)$ curve
$sigma$ proper time (massive particle) or affine parameter (massless particle, e.g. photon)
$mu = 0, 1, 2, 3$
$P^mu$ vector
If $x^mu(sigma)$ describes the curve, it may be written as
$D/dsigma P^mu = dot x^nu nabla_nu P^mu$ (1)
where:
$dot x^mu = dx^mu/dsigma$
$nabla_mu$ covariant derivative
In case of a vector, you have
$nabla_nu P^mu = partial_nu P^mu + Gamma^mu_{nu lambda} P^lambda$

Velocity
If the vector represents the position, that is $P^mu = x^mu$, (1) measures the velocity $V^mu$ of the particle
$V^mu = D/dsigma x^mu = dot x^nu nabla_nu x^mu = dot x^nu partial_nu x^mu + dot x^nu Gamma^mu_{nu lambda} x^lambda$ (2)

Acceleration
As for the acceleration $A^mu$ you apply the (1) once again to the velocity $V^mu$ in (2)
$A^mu = D/dsigma V^mu = dot x^nu nabla_nu V^mu$

Note
The covariant derivative already embeds the change of the basis vectors along a curve in a manifold to describe correctly the change of the geometric object. This is accounted for by the connection $Gamma$.
The formulas in the question do not seem correct.

“When computing the velocity of a particle moving along a curve“ I think perhaps the confusion arises from the ambiguity in the question: is the velocity of the particle itself is moving along the given curve or is the curve represents particle’s velocity (representing the change in its position ) ; does Z represent a scalar field or a vector field?