I am looking for a replacement battery for a cordless power tool.

The original is 12V 1.4Ah, I can get the same, or, 12V 2Ah and even 12V 3Ah.

Now, I understand "Ah" refers to the storage capacity of the battery, e.g. it will last longer on the same current draw, and take longer to charge fully.

However, does that mean that if the tool in question asks the battery for lots of current, the battery may be able to provide more (e.g. 3A instead of 1.4A max) ? This could damage the tool if the wiring / motor hasn't been designed to take the higher current

Thanks in advance.