I understand that each miner is working on a different block. Correct me if I'm wrong but this is because the first transaction in a block basically contains my wallet address so I can receive the bonus of solving the block. Obviously everyone wants to receive the bonus so that first transaction will be pointing to everyone's separate wallet addresses.
Assume that I have two computers solo mining with both pointing to my wallet address, wouldn't this mean my computers are solving the same block? Continuing with this train of thought, wouldn't that mean that my slower computer is basically useless because it would be generating the same hashes as my faster computer?
Now how does this apply to pooled mining? In a pooled community, is everyone solving the same block which has the first transaction pointing to the pool owner who then distributes the coins?