What was very common on those devices was using the "poke" command in BASIC to change a handful of values, but while it was possible to change code in this way it was much more common to be changing the value of variables - things corresponding to "number of lives left" and the like. Not all that different.
Fairly quickly, though, the games were entirely in machine code and used fancy loaders (still from tape mostly) so you didn't get access to BASIC. This created a market for devices that let you get at a monitor program - the "Multiface" series of addons². They had at least three generations of that device, but the company slightly weirdly evolved into a music production company³ after that, which is kind of cool.
Er, ok, I'll stop now while I still can...
Edit: PS - you should ask him about it. Tell him another former ZX81 owner says "Hi" and that my fingers still ache from that keyboard. Although I sneer a bit at its capabilities, it kicked off an interest in computing that's still paying the bills 40 something years later...
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¹ https://spectrumcomputing.co.uk/entry/2000265/Book/Not_Only_...
One could say that the difference is whether the developers intended the changes you're making to be possible or not, but what about programs with dedicated modding APIs?