What's the correct format for a phone number in Contact's "Phone Number" field?

It's the international format being used as the standard in the industry. You can find more info online by searching with relevant keywords (e.g.: 'US international phone number format') which will give you some articles, such as this Twilio article on the topic.

This is important only if you will use the SMS text messaging feature for appointment notifications in the Console, because the 3rd party service that the Console uses to send the text messages obviously requires international phone number format to be precise in their delivery.

The default interpretation of phone numbers by the delivery service is US and Canada numbers. So, if the country code is missing from a phone number, it will see it as potentially a US or Canada number and it will try to send that way. Which should probably work for actual US and Canada numbers without the country code in the number.

But delivery will fail if the numbers are not actually from US or Canada. In that case the message will show as just 'Queued' in the client's Email Log (Contacts menu > Contact Name > section 'Other Communications' > 'Status' column to the right of the message).

So, as a good practice to avoid delivery issues, it is recommended that you follow the international standard for your clients' phone numbers (in Contacts menu > Contact Name > link 'edit info' in the top right corner > Phone number fields in the Profile form), if you send them text (SMS) appointment notifications from the Console, and especially if their phone number is not a US or Canada number, but from another country.