This is what the Oxford Online dict says
play on:
exploit (a weak or vulnerable point in someone): he played on his
opponent’s nerves
play with: treat inconsiderately for one’s own amusement:
she likes to play with people’s emotions
And another reference from there shows us that to play games on someone does imply some sort of trick which can be (or not) malicious (technically). But its mostly used in a non-positive sense.
So to ans your question, both play game on and play game with can suggest the malicious nature, depending on what context its being used in. E.g:
"Quit playing games (with me) and come to the point man!"
"He is playing a game on her to like him" -Studied deception
"He will never marry her! He is just playing games with her (for sex)" - suggesting studied deception again.
play games on is less commonly used, and even lesser commonly used in British Eng compared to US, as seen in the Google Ngrams below:
The New Partridge Dictionary of Slang and Unconventional English (2008) says:
have a cow to become emotionally overwrought; to lose control US, 1966.
Speaking of Animals: A Dictionary of Animal Metaphors (1995) by Robert Allen Palmatier says:
HAVE A COW to have a cow. To have an anxiety attack. Source: COW. WNNCD: O.E. On the TV show "The Simpsons," Bart Simpson says "Don't have a cow, man!" meaning "Don't get all upset about it." Bart is likening an anxiety attack to giving birth to a cow - a frightening thought. Normally cows are the ones that give birth to cows - i.e., bull calves and heifer calves. Compare Have Kittens.
WNNCD is Webster's Ninth New Collegiate Dictionary (1983) and O.E. means Old English, but the O.E. must apply to the plain word cow rather than the phrase. (The OED dates cow to Old English.)
This Yahoo Voices article - Idioms Unpacked: "Don't Have a Cow" - also claims it means to (not) give birth to a cow, which would be distressing for a human to do. It lists a number of references at the end, but I've not followed them.
A quick search of Google Books shows this snippet dated 1962 from Field and Stream, Volume 67:
"Oh, don't have a cow," Chip said confidently. "They just haven't begun to fly yet."
"If they don't fly soon," Andy insisted, "they're going to need landing lights."
(Care must be taken with Google Books' snippets as they're often mislabelled, but following the story text we find an advert for a "NEW 1963 book of homes", so it's likely from 1962 or 1963.)
Searching Subzin.com, the first film I found to use the phrase was Sixteen Candles (1984):
00:39:00 I don't know, Jake.
00:39:02 I'm getting strange signals. Well, they're not comin' from me.
00:39:05 Everything's fine. Don't have a cow.
00:39:08 Okay.
00:39:10 Just remember one thing.
Edit: Good timing, as the OED have just released an update to the dictionary containing the phrase for the first time. The first quotation is from a 1959 newspaper:
1959 Denton (Texas) Record-Chron. 26 Mar. 3/2
He won't let me watch rock 'n roll shows... He'd
have a cow if he knew I watched 77 Sunset Strip.
Best Answer
As aparente001 points out in a comment above, the expressions tend to mean two very different things: "to catch one's breath" is a well-established idiom whose primary meaning is "to recover from exertion" (the implication being that one was "out of breath" and is now in the process of regaining one's normal breathing pattern), while "one's breath caught" means that one's normal breathing was suddenly but briefly interrupted—perhaps because one became aware of something dangerous or emotionally resonant.
Christine Ammer, The American Heritage Dictionary of Idioms, second edition (2013) has this entry for "catch one's breath":
Ammer's treatment of definition 1 of "catch one's breath" leaves room for the expression to be understood as "be excited, anxious, or nervous" (Ammer's definition of "hold one's breath" elsewhere in the same dictionary); this is not the primary sense of "catch one's breath," however, and in any case it lacks the sense of sudden, very brief, spontaneous, and (as it were) involuntary breath holding of the sort implied by "one's breath caught."
Nevertheless, a search of Google Books matches for "caught [one's] breath" does turn up a number of instances in which the expression seems interchangeable with "[one's] breath caught" in the sense of a stifled gasp. For example, from Fanny Kilbourne, "Betty Bell and the Leading Man," in The Delineator (January 1920):
And from Stella Gammell, The Immortal Throne (2016):
The expression "[one's] breath caught" has become remarkably more frequent in Ngram search results over the past half century—and especially since about 2000—than it was before that, as this Ngram chart for "breath caught" for the period 1850–2019 indicates:
Although relatively uncommon in the nineteenth century, the expression "breath caught" in the relevant sense does appear at least as early as 1817. From Grace Keon, The Ruby Cross: A Novel (1817):
Modern instances of the expression tend to describe a similar response to unexpected or intense emotion. For example, from KJ Jackson, The Iron Earl: Valor of Vinehill (2019):
To judge from Google Books search results, the wording has become almost a cliché in Harlequin Romance–type novels. Indeed, the sentence "Her breath caught in her throat" appears twice (many pages apart) in The Iron Earl.
Instances of "caught [one's] breath" may still be more common than instances of "[one's] breath caught," as this Ngram chart comparing the frequency of occurrence of "breath caught" with that of "caught her breath," "caught his breath," "caught my breath," "caught their breath," and "caught our breath" for the period 1850–2019 suggests:
But in any event the comparison is not apples to apples, since so many of the "caught [one's] breath" instances involve regaining one's breath, not momentarily losing it.