I am not sure whether there is such a thing as a 'standard'. I have several poles from different US and European manufacturers. The spikes are all truncated cones, with a starting diameter of 11mm increasing to 12mm or so. That includes my BD poles. I also have a selection of 'rubber' tips from BD, Forclaz and some others. I haven't been able to find a 12 mm bore diameter amongst them. Most are 11 mm in diameter, ensuring a good friction fit. The Forclaz tip is interesting, as the bore isn't smooth, but has a series of ridges around the inside. The lands have an internal diameter of 11mm, the grooves are 12mm.
Nonetheless,
@t2andreo's suggestion to tape the end of the pole is an excellent one if the tips are not a good friction fit. I have also used this approach on the other 'standard' diameter spike used by BD and Komperdell and maybe others. BD Z poles and some Komperdell poles have a spike that starts at about 9 mm. For some time in Oz, it wasn't possible to get the BD or any other tips in this internal diameter, and I taped up the end of my wife's Z poles so she could use a 11 mm tip.
I find it rare these days to see tips that don't have an internal washer to prevent the spike pushing through, although I have seem some in the past. Clearly worth checking if you are going to buy cheap tips along the way. It is possible to recover the washer from a worn out tip with not too much effort and a sharp knife. Cut open the tip and it should be a relatively simple matter to extract the washer. Or buy better quality tips and carry them. They are not heavy!
There is an adjustment mechanism in the middle section of the poles that can shorten or lengthen the core running down the pole that would address this. It sounds like you would need to shorten the core until the button just activates with a gentle pull. It doesn't need to be too tight, but enough to keep the button in contact when the pole is extended. BD have the
instruction sheet on line, with a more complete explanation about how to do this.