Can I replace the original laptop battery with an OEM battery with higher voltage / power?
Over the years, the maximum charging cut-off voltage for newer Li-ion batteries has gone up a bit, because of improvements and perhaps a bit of marketing.
The three cells in a (nominally) 11.1V battery pack are usually charged to 4.2V per cell, the (newer and slightly different) cells in a (nominally) 11.4V battery pack are usually charged to 4.35V per cell.
Replacing a 11.4V battery pack with an 11.1V battery pack will cause trouble because the cells will be overcharged.
Replacing a 11.1V battery pack with a 11.4V battery pack should be OK, but the cells will not be fully charged so you don't get to use their quoted full capacity. On the up side: they will last a bit longer (more charging cycles).
The higher top voltage translates into a slightly higher average voltage over a discharge cycle, but it is very unlikely this will bother your laptop.
Having said that, the warnings in the comments do apply, and to be sure, you should get either a different battery pack or a very good guarantee.