This is taken from a site similar to this:
"Intention is the general word implying a having something in mind as a plan or design, or referring to the plan had in mind. Intent, a somewhat formal term now largely in legal usage, connotes more deliberation (assault with an intent to kill)"
Thus, an intention is a general plan that one has in mind. An intent is really a deliberate plan.
I use just very often in speech and casual writing. I tend to edit it out of more formal writing. I am a native American English speaker from Tampa and Boston.
To me, it can have both meanings, and I don't think that I have a preference. Your examples are all ambiguous to me. I think that the meaning is often clear from context (don't underestimate the power of context!), as in:
1) I'll be there soon. I just woke up. (recently)
2) I didn't call her. I just sent her an email. (merely)
3) I tried, but I just don't understand. (simply)
I don't see how changing tenses or aspects can help you, except that the future and simple present (always?) rule out the recently interpretation. But all of the following are equally ambiguous to me (sans disambiguating context).
4) I [just ate/am just eating/was just eating/have just eaten/etc.] an apple.
When I want to be crystal clear, I just replace just with merely, only, simply, or suchlike. I think that I less often change just to recently. I think I more often say just now to indicate that I mean recently.
Best Answer
'With the purpose' implies that the thing you are doing is contributing to your goal. 'With the intention' only implies that you had another goal in mind. The action doesn't have to contribute to it.
So you can say
The walk doesn't contribute to the studies, so you could not substitute ' purpose' for 'intention'. If ' purpose' is correct you can usually substitute 'Intention'.