Which Masi model is this?

Very hard to tell - Masi have used a relatively "traditional" frame design.

Masi made a bunch of frames with trackends for the fixed/single speed market, so theres a chance this was one-speed from new. The pedals don't have foot retention, so I'd be leaning to originally a single speed rather than fixed gear.

The visible components aren't showing anything that denotes an age.

That it has cross interruptor brake levers on drop bars might be a later addition, or the "track" style of large radius drop bar might have been in the original design.

Best I got is "built in the last 20 years" as a road single speed bike.

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Frankly it looks pretty nice condition. If it fits you, and you find it comfortable, then the model is irrelevant, and the age really doesn't matter. Instead, look at the wear of the chain, chainring, and cog as an indicator of overall wear.

It looks a lot like a Masi Speciale Fixed LTD

This is a picture of a 2010 model

Link to the 2010 catalog
Link to the Masi catalog page