The link you posted is about time-invariance of the dynamical system, meaning that the behavior of the system at some time $t$ is the same as at some other time $t′$.
For instance, the dynamical system $\dot{x}(t)=ax(t)$ where $a\in\mathbb{R}$ is time invariant. If you start the system at time $t_0$ from the state $x(t_0)=x_0$ and if you start the system at time $t_1\ne t_0$ from the state $x(t_1)=x_0$, then the two trajectories will be translated versions of each other.
The dynamical system $\dot{x}(t)=tx(t)$ is not time-invariant, but time-varying. In this case, the initial time will matter.
The composition formula
$${\displaystyle \Phi (t_{2},\Phi (t_{1},x))=\Phi (t_{2}+t_{1},x),}$$
works for both time-varying and time-invariant systems (with no input).
Finally, the example you gave is indeed a dynamical system whose state is given by $x(t)=x(t_0)+(t-t_0)^2/2$. In this case, the system has an input and this is not captured by the $\Phi$ formula you gave which is for systems with no input. In fact, the solution to that system is given by
$$x(t)=\Psi(t,s)x(s)+\int_s^t\Psi(t,\theta)\theta d\theta,$$
where $\Phi(t,s,x)=\Psi(t,s)x=x$, $t\ge s$, and where $\Psi(t,s)$ is the so-called state transition matrix (see e.g. https://en.wikipedia.org/wiki/State-transition_matrix). Finally, since the system is time-varying, the trajectory will not only depend on the initial condition but also on the initial time.
If you want to know more about dynamical systems, you may look at "Nonlinear Dynamics and Chaos" by Steven Strogatz. It is usually a good starting point.
Best Answer
My impression is that, in dynamic systems theory, the two terms 'orbit' and 'trajectory' are often used interchangeably. They both seem to come from physics, e.g. orbits of planets, etc. Earlier, I had the impression that 'trajectory' was more commonly used when talking about continuous systems, and that 'orbit' was used for discrete ones. This distinction does not seem to be even nearly universally accepted however, and is probably more of a preference than anything else. Possibly this impression comes from the fact that you use 'orbit' in group theory, when a group is acting on elements of a set, see definition here. It therefore feels somewhat more natural to call the sequence emerging from a discrete system, orbit. Trajectory seems to be somewhat more common among physicists. I get a feeling that this is similar to the distinction (or non-distinction) between 'map' and 'function', see discussion here.
I find it hard to find any real distinction being made explicitly in dynamical systems literature (at least in the many books on my shelf). Three examples of the terms being explicitly used interchangeably by respected authorities are:
I want to make clear that I am a mathematician and not a physicist, therefore my impression is very much based on literature about pure and applied mathematics. It might be the case that physicists make more of a distinction between the two, even though I doubt it.
Another, perhaps more important distinction that should be made when talking about orbits/trajectories, is whether one means orbit/trajectory a as defined in 1.1.1, as a sequence, or as a set. Often this does not pose any problems, as it is clear from context. Personally, dealing mostly with discrete systems, I usually write $\{f^n\}_{n\geq 0}$, with set braces, for the unordered set, which could be finite for a periodic orbit, and $(f^n)_{n\geq 0}$ for the ordered infinite sequence. This is somewhat OT, but nevertheless important to have in mind when reading about these things.
I hope that this either suffice as an answer, or at least makes things a bit more clear.