Enter, the Function Monkey ...
As described in the link, to plot a graph, simply imagine the $x$-axis covered in coconuts, one for every $x$ value, like this:
$\cdots \quad (-3\;) \quad \color{#9509A5}{(-2\;)} \quad \color{green}{(\;-1\;)} \quad \color{red}{[\;0\;]} \quad \color{ #2B87CD}{(\;1\;)} \color{#D86907}{\quad (\;2\;)} \quad (\;3\;) \quad \cdots$
with "$\color{red}{[\;\cdot\;]}$" indicating the coconut at the origin. The Function Monkey strolls along the axis, picks up each $x$ coconut, evaluates the corresponding $y$ value (as is his wont), and throws the coconut to the appropriate height (or depth). Graph plotted!
But, wait! No one ever said that the values of the coconuts were required to match the values of the $x$ coordinates which have been coloured for convenience. Hmmmm ...
If the coconut at (the coloured) location $x$ has value $x+1$, then the row of coconuts looks like this: $\cdots \quad (-2\;) \quad \color{#9509A5}{(\;-1\;)} \quad \color{green}{(\;0\;)} \quad \color{red}{[\;1\;]} \quad \color{ #2B87CD}{(\;2\;)} \color{#D86907}{\quad (\;3\;)} \quad (\;4\;) \quad \cdots$
Importantly, the coconut locations in colour have not changed. The coconuts still sit on the axis at locations $x = \cdots, -3, \color{#9509A5}{-2},\color{green}{-1}, \color{red}{0},\color{#2B87CD}{1}, \color{#D86907}{2}, 3, \cdots$ still indicating the coconut at the origin.
So far as the Function Monkey is concerned, adding $1$ to each coconut value has effectively shifted the row of coconuts to the left. (Likewise, adding $-1$ to each coconut value effectively shifts the row of coconuts to the right.) Since the Function Monkey throws coconuts vertically, the plotted graph shifts the same way.
In a similar manner, multiplying each (original) coconut value by $2$ yields this row of coconuts:
$\cdots \quad (-6\;) \quad \color{#9509A5}{(-4\;)} \quad \color{green}{(-2\;)} \quad \color{red}{[\;0\;]} \quad \color{ #2B87CD}{(\;2\;)} \color{#D86907}{\quad (\;4\;)} \quad (\;6\;) \quad \cdots$
Again, the locations of the coconuts haven't changed, but we see that the span of coconut values from $-6$ to $6$ has been compressed into the space between locations $x=-3$ and $x=3$. The graph will exhibit the same, horizontal, compression.
A plot-less way of thinking about this, which is close to your "head start" interpretation, is that "$f(x+1)$" fools Function Monkey $f$ into thinking that each input value is one unit bigger than it really is. Similarly "$f(2x)$" fools him into thinking that each input is twice its actual size. The effective changes in output values correspond to the Function Monkey's altered perceptions.