I think that this question has been already covered on the site. I'm answering via an example. Both your citations are correct.
X
-.910218691 -.037212328 -.275548166 -.360441580
-1.855949714 -1.767057411 -.317283890 1.633887711
-.186585570 -.316960442 -.340387838 -1.160337400
1.427254188 -.872044568 -.447456104 -1.785203938
-1.524117328 1.013061127 .735027680 .583921184
.014211669 -.880275500 1.516716199 -.266348030
svd(x) = UDV'
D
3.781503197 .000000000 .000000000 .000000000
.000000000 2.427179503 .000000000 .000000000
.000000000 .000000000 1.762938484 .000000000
.000000000 .000000000 .000000000 1.494400129
.000000000 .000000000 .000000000 .000000000
.000000000 .000000000 .000000000 .000000000
U
-.1045359966 .0613688431 .1533784675 .5925648676 -.7620680911 -.1729168539
-.6655219390 .7063271894 .1384176823 -.0473139686 .1623946521 -.1020409758
.1754941747 .1964628810 .0893095485 .6365658833 .3639134791 .6204665918
.5934543082 .4282595430 .0031973635 .1744522991 .2294862700 -.6174932972
-.4039262123 -.4669763619 -.2167568310 .4571230653 .4262068341 -.4256517430
-.0031887094 .2393009491 -.9499187589 .0449114759 -.1616507584 .1105948958
V
.7299152783 -.0317492361 -.0520390915 -.6808139286
.0529861206 -.9763855955 .1901410033 .0878068268
-.1023534922 -.1976805415 -.9745608685 -.0260245989
-.6737506364 -.0811514539 .1066276016 -.7267083844
UDV' restores X
-.910218691 -.037212328 -.275548166 -.360441580
-1.855949714 -1.767057411 -.317283890 1.633887711
-.186585570 -.316960442 -.340387838 -1.160337400
1.427254188 -.872044568 -.447456104 -1.785203938
-1.524117328 1.013061127 .735027680 .583921184
.014211669 -.880275500 1.516716199 -.266348030
-----------------
Observe that (because X wasn't square) D above is not square.
The singular values are only four, the min(nrows,ncols) in X.
So, cut empty rows (or columns) in D so it becomes square, 4 x 4.
D
3.781503197 .000000000 .000000000 .000000000
.000000000 2.427179503 .000000000 .000000000
.000000000 .000000000 1.762938484 .000000000
.000000000 .000000000 .000000000 1.494400129
Then leave only 4 first columns ("real" left eigenvectors) in U
U
-.1045359966 .0613688431 .1533784675 .5925648676
-.6655219390 .7063271894 .1384176823 -.0473139686
.1754941747 .1964628810 .0893095485 .6365658833
.5934543082 .4282595430 .0031973635 .1744522991
-.4039262123 -.4669763619 -.2167568310 .4571230653
-.0031887094 .2393009491 -.9499187589 .0449114759
And only 4 first columns ("real" right eigenvectors) in V
(in this example, V was initially 4-column, so leave as is)
V
.7299152783 -.0317492361 -.0520390915 -.6808139286
.0529861206 -.9763855955 .1901410033 .0878068268
-.1023534922 -.1976805415 -.9745608685 -.0260245989
-.6737506364 -.0811514539 .1066276016 -.7267083844
UDV' restores X as well as it was above
-.910218691 -.037212328 -.275548166 -.360441580
-1.855949714 -1.767057411 -.317283890 1.633887711
-.186585570 -.316960442 -.340387838 -1.160337400
1.427254188 -.872044568 -.447456104 -1.785203938
-1.524117328 1.013061127 .735027680 .583921184
.014211669 -.880275500 1.516716199 -.266348030