Cheap umbrellas are cheap for a reason. They typically do not have the nice thick stem walls that the more expensive ones do and so are prone to crushing were you tighten them to a flash head. They also do not usually have the nicely chamfered end for easy insertion into the flash head mounting.
There are two common umbrellas stem diameters. There is the 8mm (largely universal and will fit just about any head) and there is the 7mm (used by Elinchrom). Obviously the 8mm will not fit the Elinchrom 7mm mounting. If you invest in Elinchrom flash heads (which are very good) then be prepared to also invest in a range of 7mm umbrellas.
The manufactures own umbrellas tend to be expensive. This is not unreasonable as they are typically made to a much higher standard than the cheap ones available from Ebay for example. However, the quality of light from either is very similar. To avoid or reduce the chances of crushing the wall on the umbrella stem you need some 7mm dowel (6mm for Elinchrom). As the cheap umbrellas rarely come with a fancy chamfered end it is simply a matter of running a bit of general purpose glue down the stem and then pushing in the dowel until it will go no further and then pull it back a 1/4 inch. Cut off the excess dowel and job done. You may wish to lightly file the end of the stem to make it more rounded and easier to fit into the flash mounting whilst you are in DIY mode.
All umbrellas tend to protrude from the flash mount and are usually at a height just right to take some ones eye out. Drill a 5mm hole in a wine bottle cork about 2cm deep and fit to the end of the umbrella shaft. If some one does catch themselves on the umbrella shaft you will do far less damage.