How do twitch effects work with the stack vs activated tap abilities?
There are certain actions that you undertake in Magic which do not use the stack, and therefore cannot be responded to. (AFAIK this has never been different for as long as the stack has existed, so this is not really a recent change.) The most important of these are:


Paying costs
Activating mana abilities


Your first example above is an instance of #1. The cost of the Steel Overseer's ability is to tap the Overseer. After the Overseer is tapped, the +1/+1 ability goes on the stack, and each player gets priority. At this time it's too late to tap the Overseer, since it's already tapped and the ability is already on the stack.

Your second example is an instance of #2. Tapping a land for mana does not use the stack,  so you cannot attempt to tap a land in response to your opponent starting to play a spell. By the time you receive priority, the land is already tapped and the mana is in your opponent's pool (or has already been spent to put the spell on the stack).

Generally, then, you need to play your tap abilities before your opponent has the chance to do anything. This means:


If you're trying to deprive your opponent of mana, you should tap his lands in his upkeep step. Your opponent will be able to tap them in response to pay for instant-speed spells and abilities if he has any, but he won't be able to use that mana for any main-phase spells.
If you're trying to keep a creature from attacking, the last time when you can do so is the Begin Combat step. By the time you get priority in the Declare Attackers step, your opponent's creatures are already tapped and attacking, so tapping them has no effect.
If you're trying to prevent your opponent from using some sort of other tap ability on a creature or artifact, your best bet is to tap it at a time when the ability is not relevant or has no targets. For example, tap your opponent's Avacynian Priest in his end step. He could activate the ability in response, but it doesn't matter since your creatures will untap at the beginning of your turn anyway.
If you're trying to tap something like the Steel Overseer, which has an ability that's always relevant, you're basically SOL.