We need to check the reverse : that a payment method has been selected and that the basket is marked : pending, completed, processed or whatever

Such a state doesn't exist. The basket isn't updated when clicking the payment button. It's only updated after the payment processor has done something (e.g. payment is coming through). A pending basket means the basket has been processed and payment from the processor is pending. For 99% of payment processors it will never go payment since card payments are instant.

If we check onCPayAfterPaymentBasketUpdated Not Egal To Notinitiated, we would receive an email at each different state ?

You'll receive an email anytime that trigger fires and the basket payment_status is set to NotInitiated. I've no idea how many times that might be. You'll need to test that.

So how can we trigger only one auto action (for emailing) when onCPayAfterPaymentBasketUpdated Not Egal To Notinitiated and Basket is either Pending, processed, completed ?

For offline payments it can be in a pending state. For other gateways it likely will never go pending. You can probably have 2 auto actions to cover both scenarios. 1 for offline payments and 1 for card payments. Example conditions for these actions as follows.

Action 1 (Offline) Conditions
1: [var1_payment_status] Equal To Pending
2: [var1_gateway_account] Equal To OFFLINE_GATEWAY_ID_HERE

Action 2 (Cards) Conditions
1: [var1_payment_status] Equal To Completed
2: [var1_gateway_account] Equal To CARD_GATEWAY_ID_HERE

If we want to cover two gateways is it possible to separate them with a | like :
[var1_gateway_account] Equal To OFFLINE_GATEWAY_ID_1|OFFLINE_GATEWAY_ID_2

Yes, but you need to change the usage to REGEXP as follows.

[var1_gateway_account] Is REGEX /ID_1|ID_2/