It doesnt mean that the language is misleading but rather that the natural parsing of the language is wrong.

I.e. we naturally parse this sentence as "(Texas) Appeals (Court Rules Phone Search)" which is wrong. The correct parse is "(Texas Appeals Court) Rules (Phone Search)". I.e. at first glance we think that "appeals" is a verb rather than an adjective.

Unambiguous headline:

Texan Appellate Court Rules That Searching the Telephones of Arrestees Violates 4th Amendment Protections

For the first example to parse, wouldn't it need to be "(Texas) Appeals (Court's Rule of phone search)..." or something along those lines? That is, the grouping doesn't parse correctly without at least a possessive apostrophe which the Title lacks. I'm failing to detect a correct parsing under the grouping you offer. I'm not sure that a parsing path that requires parsing errors can be considered the more likely for the reader to take. In my case, I parsed it correctly from the beginning.

You're missing the point. It's not that the sentence has to have multiple logical interpretations; it's that your brain begins to parse it one way and then comes to words that don't fit that interpretation. I had to jump back to the beginning twice because I assumed "appeals" was the verb at first.

OK, that makes sense. I guess then I'm not sure whether this was meant as a criticism of the title or just pointing out the feature. Since "Texas Appeals Court" is a single entity, it seams like a reasonable way to begin the headline, especially given the general convention that headlines avoid unnecessary prepositions such as in the possible alternative "Appeals Court in Texas". To my reading, the garden path ends at the third word, which doesn't seem too terribly bad for a headline. Perhaps I'm missing some subtle annoyance though.

I parsed it correctly because I'm used to reading about appeals courts ruling on things, but one could take "rules" as a noun and "phone" as a verb (the court's rules telephoned a search to let the search know that it was illegal?).

"Search" could also be a verb, but that makes it more of a stretch (even if we envision a kind of phone called a "rules phone", there would be a singular/plural agreement problem in taking it as the subject of "search").

Exactly -- this is just the context of the word rules; when you read legal stuff you know to go instantly to the other meaning.