Hmm, the phrases in question are indeed very thinly nuanced.
Simple explanation would be the closest translation of Russian words to English:
- лишь - just
- только - only
- просто - simply
"Лишь" is associated with minimum effort or cost.
"Только" is associated with common sense limitations.
"Просто" is associated with minimum complexity.
All meanings suggest the lack of hidden agenda and that the other party should not draw any conclusions.
P.S. Even with "Лишь" being translated as "Just", the phrase "I tried just ..." is perfectly useful, common, and apologetic, while "Я старалась лишь ..." in Russian sounds more bookish and rather dismissive.