Russian speakers would often ignore the differences and use all those words interchangeably indeed. You're quite safe combining them randomly, but should be aware of the reasons of such disregard.
The knowledge of agriculture, nature and geology has dropped significantly. In big cities many people couldn't explain how potato grows and simply wouldn't survive in the country without food delivered to stores. Some can't explain the difference between a "rock" and a "mountain". You shouldn't expect that everybody actually understand what
луг (meadow) or
поле (field) precisely mean. They might just misusing it.
Many Russian people love to toy (tinker, fool) around with the words. We often do it just for fun. One can deliberately use an archaic form of a word, a synonym in the wrong context, a metonymy, etc. Thus if a person uses the phrase: "вчера подстриг свою лужайку", it is entirely possible that he is aware of the word's true meaning (one doesn't trim a wild meadow, mind you) and is just being casual with you.
When you participate in a formal conversation, you should familiarize with the real, proper meanings of those words, lest you appear uneducated or inadequate.
Газон - can be literally translated as "lawn". It's the safest, default, straightforward option for a lawn-mowing discussion. (I disagree with the @AR. answer about this word's usage frequency and modern habits; in my urban experience it's a basic, very common word. To educated Russians it means any plot of seeded soil, regularly maintained for aesthetical reasons, often with flowers. It doesn't matter who in particular is responsible for the maintenance; it might be personal or public.)
Трава - is the "grass", all and any of it. It's a very general, broad and therefore flexible word. Use it with discretion, as you would go about any abstract concept.
Трава is less dull than
Лужайка - the hypocorism (!) of the
луг, i.e. "meadow". One normally can't confuse a maintained, confined lawn with a vast, uneven, overgrown area outside anyboby's property. Avoid using it in the meaning of "lawn" in the formal speech. The hypocorism version
лужайка is even less appropriate, as it is considered funny. It might even be recognized as an obscene joke - this hypocorism is commonly used as a slang for "pubic hairs". It could also be used to jokingly exaggerate the lawn size of a rich person (or humiliate if the lawn is laughably small). This option is quite risky for unprepared wielder.
Стричь - a straightforward version. It litterally means "trim" and is widely used.
Косить - also widely used, usually preferred, but comes with a subtle notion (rarely seen by anybody). Semantically it means: "detach and preserve the surface parts of a grass for agricultural use". Harvesting crops, basically. Note that trimming of a decorative lawn usually pursues the goal of removing (shortening) the herbs; the removed top part might get reduced to a pulp in the process. This is totally unacceptable when it comes to "mowing" and "harvesting". People who aren't affiliated with agricultural business don't care. Some country people would probably shun from using
косить in the lawn-mowing discussion themselves. Overall it is safe to use.
- Стричь газон
- Стричь траву
- Стричь лужайку (hee-hee)
- Косить траву
Other pairs sound strange to me:
- Косить газон
- Косить лужайку (the speaker likely says about harvesting a meadow for agricultural use)