battery low voltage cut off breaker (manually reset)
You identified the problem very precisely: a relay hardly has a specific &quot;turn-off voltage&quot;. Your idea with the transistor is great: You can build circuits with a very precisely defined voltage-dependent behaviour from semiconductor components like that.
Things get even better: one of the very big downsides of your relay approach is that keeping the relay closed will inevitably consume quite a bit of current (to keep the coil energized), which will drain your battery (and hence work against the purpose of the battery).
A MOSFET (Metal-Oxide-Semiconductor Field-Effect Transistor) is cool in that it changes the resistance of its junction (the connection between the contacts called &quot;drain&quot; and &quot;source&quot;) purely based on the voltage applied to its gate contact – and potentially, quite drastically so!
Even a single P-channel MOSFET in place where your relay's switch used to be, together with a  voltage divider made from very large resistor values (to avoid draining the battery) could solve this. Better, yet, there's voltage reference-comparators like TL431LI that are meant for exactly these kinds of things, switching a transistor at exactly the right input voltage on or off. I'd frankly look up &quot;high-side switch MOSFET&quot; circuits, and marry one to a &quot;TL431 undervoltage&quot; detector.
However, the solution gets even easier: there's undervoltage protection circuits against over-discharging batteries. Sounds like exactly the job you have. Look up TI's BQ296100 and chips in the same category if this spikes your interest.