Does 'karma' mean that everything that happens to us is under our control, or only that we are responsible for it?
Not everything is caused by karma. Some things are caused by other reasons, like the weather. The Sivaka Sutta talks about this. There's a summary at the bottom:

Bile, phlegm, and also wind,
Imbalance and climate too,
Carelessness and assault,
With kamma result as the eighth.

To quote the sutta in detail:

“Some feelings, Sīvaka, arise here originating from phlegm disorders …
originating from wind disorders … originating from an imbalance of the
three … produced by change of climate … produced by careless behaviour
… caused by assault … produced as the result of kamma: that some
feelings arise here produced as the result of kamma one can know for
oneself, and that is considered to be true in the world. Now when
those ascetics and brahmins hold such a doctrine and view as this,
‘Whatever a person experiences, whether it be pleasant or painful or
neither-painful-nor-pleasant, all that is caused by what was done in
the past,’ they overshoot what one knows by oneself and they
overshoot what is considered to be true in the world. Therefore I say
that this is wrong on the part of those ascetics and brahmins.”

The Acintita Sutta states that you can't work out karma and its results precisely:

&quot;The [precise working out of the] results of kamma is an
unconjecturable that is not to be conjectured about, that would bring
madness &amp; vexation to anyone who conjectured about it.

What's more skillful is to develop one's virtue (sila), concentration (samadhi) and wisdom (pañña) as stated in the Lonaphala Sutta. That will mitigate the ill effects of past karma (actions).
As for future karma (actions), to avoid accumulating karma debts, it is skillful to adopt the following skillful thinking from the Themes Sutta:

“And for the sake of what benefit should a woman or a man, a
householder or one gone forth, often reflect thus: ‘I am the owner of
my kamma, the heir of my kamma; I have kamma as my origin, kamma as my
relative, kamma as my resort; I will be the heir of whatever kamma,
good or bad, that I do’? People engage in misconduct by body, speech,
and mind. But when one often reflects upon this theme, such misconduct
is either completely abandoned or diminished. It is for the sake of
this benefit that a woman or a man, a householder or one gone forth,
should often reflect thus: ‘I am the owner of my kamma, the heir of my
kamma; I have kamma as my origin, kamma as my relative, kamma as my
resort; I will be the heir of whatever kamma, good or bad, that I do.’