We weren't even allowed to do that, in case family members (or whoever) found out that an NHS person had called. Clearly policies do differ somewhat, and it sounds like you have a more useful compromise ;-)
The above section details types of phrasing to avoid; however, it doesn’t detail what users should NOT say on their greeting. Though this is a bit loaded, as there are hundreds of combinations of things one shouldn’t say, there are some key components users should ALWAYS avoid. a. Forget About Slang: You should strive to be as professional and welcoming as possible in your greeting. While this may steer you towards using slang, in an attempt to make callers comfortable, it’ll most likely work against you. As a professional, your demeanor, tone, and speech should be clear cut and well articulated. Using slang undercuts this and works against you. b. Don’t Even Think About Profanity: This is a no-brainer. Never, under any circumstances, curse in your greeting EVER! c. Keep Your Sentences Clean, Don’t Ramble: Introduce yourself and give your caller specific direction. Avoid long diatribes detailing tangent thoughts. Keep it simple and quick. d. Always Return Your Calls: It’s important for callers to feel they are valued. Nothing dissolves this quicker than a greeting that doesn’t stress this. For example, “I’ll call you when I can,” “If I don’t return your call, please call back”—these phrases are terrible and completely destroy any good will you may have with a caller.

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