You appear to be assuming that the largest eigenvalue necessarily can be paired with the largest coefficient within the eigenvectors. That would be wrong.
The question clearly transcends software choice. Here is a fairly silly PCA on five measures of car size using Stata's auto dataset. I used a correlation matrix as starting point, the only sensible option given quite different units of measurement.
. pca headroom trunk weight length displacement
Principal components/correlation Number of obs = 74
Number of comp. = 5
Trace = 5
Rotation: (unrotated = principal) Rho = 1.0000
--------------------------------------------------------------------------
Component | Eigenvalue Difference Proportion Cumulative
-------------+------------------------------------------------------------
Comp1 | 3.76201 3.026 0.7524 0.7524
Comp2 | .736006 .427915 0.1472 0.8996
Comp3 | .308091 .155465 0.0616 0.9612
Comp4 | .152627 .111357 0.0305 0.9917
Comp5 | .0412693 . 0.0083 1.0000
--------------------------------------------------------------------------
Principal components (eigenvectors)
----------------------------------------------------------------
Variable | Comp1 Comp2 Comp3 Comp4 Comp5
-------------+--------------------------------------------------
headroom | 0.3587 0.7640 0.5224 -0.1209 0.0130
trunk | 0.4334 0.3665 -0.7676 0.2914 0.0612
weight | 0.4842 -0.3329 0.0737 -0.2669 0.7603
length | 0.4863 -0.2372 -0.1050 -0.5745 -0.6051
displacement | 0.4610 -0.3390 0.3484 0.7065 -0.2279
----------------------------------------------------------------
(Output truncated.)
The first component picks up on the fact that as all variables are measures of size, they are well correlated. So to first approximation the coefficients are equal; that's to be expected when all the variables hang together. The remaining components in effect pick up the idiosyncratic contribution of each of the original variables. That is not inevitable, but it works out quite simply for this example. But, to your point, you can see that the largest coefficients, say those above 0.7 in absolute value, are associated with components 2 to 5. There is nothing to stop the largest coefficient being associated with the last component.
(UPDATE) The eigenvectors are informative, but it is also helpful to calculate the components themselves as new variables and then look at their correlations with the original variables. Here they are:
pc1 pc2 pc3 pc4 pc5
headroom 0.6958 0.6554 0.2900 -0.0472 0.0026
trunk 0.8405 0.3144 -0.4261 0.1138 0.0124
weight 0.9392 -0.2856 0.0409 -0.1043 0.1545
length 0.9432 -0.2035 -0.0583 -0.2245 -0.1229
displacement 0.8942 -0.2909 0.1934 0.2760 -0.0463
Here trunk is the variable most strongly correlated with pc3, but negatively. A story on why that happens would depend on looking at the data and the PCs. I don't care enough about the example to do that here, but it would be good practice.
Although I produced the example with a little prior thought, and it is suitable for your question, and it is based on real data, it is also salutary: interpreting the PCs may be no easier than interpreting something more direct such as scatter plots and correlations. However, not every PCA application depends on ability to interpret PCs as having substantive meaning, and much of the literature warns against doing that in any case. For some purposes, the whole point is a mechanistic reordering of the information in the data.
You get the coefficients from PCA. These coefficients are multiplied by your observation matrix to obtain the components. So, multiply rotation by the new observation matrix instead. Don't forget to center it.
Here's the code.
Run PCA and see how the score matrix is obtained from the original data and the rotation. Note, that I'm NOT centering, and you probably should.
> x=matrix(c(1,2,3,2,4,5.5),3,2)
> x
[,1] [,2]
[1,] 1 2.0
[2,] 2 4.0
[3,] 3 5.5
> r=prcomp(x,retx=1,center=FALSE)
> r$rotation
PC1 PC2
[1,] -0.4666132 0.8844615
[2,] -0.8844615 -0.4666132
> r$x
PC1 PC2
[1,] -2.235536 -0.04876479
[2,] -4.471072 -0.09752958
[3,] -6.264378 0.08701220
> x %*% r$rotation
PC1 PC2
[1,] -2.235536 -0.04876479
[2,] -4.471072 -0.09752958
[3,] -6.264378 0.08701220
Now, apply the same rotation to the different data (again, see that I am NOT centering).
> y=matrix(c(1,2,3,2,4,6.5),3,2)
> y
[,1] [,2]
[1,] 1 2.0
[2,] 2 4.0
[3,] 3 6.5
> y %*% r$rotation
PC1 PC2
[1,] -2.235536 -0.04876479
[2,] -4.471072 -0.09752958
[3,] -7.148839 -0.37960095
Note the similarity of the new scores.
By the way, this is used a lot in forecasting with PCA. We obtain the rotation on historical data, then apply it to new data.
Best Answer
You are absolutely correct in observing that even though $\mathbf{u}$ (one of the eigenvectors of the covariance matrix, e.g. the first one) and $\mathbf{X}\mathbf{u}$ (projection of the data onto the 1-dimensional subspace spanned by $\mathbf{u}$) are two different things, both of them are often called "principal component", sometimes even in the same text.
In most cases it is clear from the context what exactly is meant. In some rare cases, however, it can indeed be quite confusing, e.g. when some related techniques (such as sparse PCA or CCA) are discussed, where different directions $\mathbf{u}_i$ do not have to be orthogonal. In this case a statement like "components are orthogonal" has very different meanings depending on whether it refers to axes or projections.
I would advocate calling $\mathbf{u}$ a "principal axis" or a "principal direction", and $\mathbf{X}\mathbf{u}$ a "principal component".
I have also seen $\mathbf u$ called "principal component vector".
I should mention that the alternative convention is to call $\mathbf u$ "principal component" and $\mathbf{Xu}$ "principal component scores".
Summary of the two conventions:
$$\begin{array}{c|c|c} & \text{Convention 1} & \text{Convention 2} \\ \hline \mathbf u & \begin{cases}\text{principal axis}\\ \text{principal direction}\\ \text{principal component vector}\end{cases} & \text{principal component} \\ \mathbf{Xu} & \text{principal component} & \text{principal component scores} \end{array}$$
Note: Only eigenvectors of the covariance matrix corresponding to non-zero eigenvalues can be called principal directions/components. If the covariance matrix is low rank, it will have one or more zero eigenvalues; corresponding eigenvectors (and corresponding projections that are constant zero) should not be called principal directions/components. See some discussion in my answer here.