Does ASIC miners leave a trace in the mining history of bitcoin? I have a research interest in understanding the spread of innovation, and the adoption of ASIC machines in the Bitcoin mining community is particularly relevant.

Does ASIC machines typically leave a trace in the blocks that they mined? Would a particular type of machine leave such trace/record/identifier in the transaction record?
No. ASICs do only hash calculation. All data in block (except nonce) is defined by pool/node software