The proper definition of an (unbroken) conformal field theory is that all scalar n-point functions (or equivalently, their generating functional, a kind of partition function) remain unchanged when each field (including the metric, if it is a field) transforms according to a representation of the conformal group and the volume element (if the metric is not a field) is transformed by the standard conformal factor.

All other statements you mention are proxies for this, equivalent under additional assumptions only. For example, since you asked about conformal invariance but stated things about scale invariance only, you always need to add the requirement of Poincare co/invariance of the respective fields. Apart from that, you need to assume a specific field contents, and for 3. also that the CFT is described by an action (many are not). 2. and 4. must be assumed together.

Condition 1. is a consequence of the fact that CFTs are fixed points of the renormalization group flow. Conversely, fixed points are scale invariant, but it is an open question whether that implies conformal invariance. But scale invariant QFTs are typically conformal.

In any case, for scaling the physical (i.e., renormalized) fields, $\Delta$ must be the true dimension, including anomalous terms (if there are any).

The proper definition of an (unbroken) conformal field theory is that all scalar n-point functions (or equivalently, their generating functional, a kind of partition function) remain unchanged when each field (including the metric, if it is a field) transforms according to a representation of the conformal group and the volume element (if the metric is not a field) is transformed by the standard conformal factor.

All other statements you mention are proxies for this, equivalent under additional assumptions only. For example, since you asked about conformal invariance but stated things about scale invariance only, you always need to add the requirement of Poincare co/invariance of the respective fields. Apart from that, you need to assume a specific field contents, and for 3. also that the CFT is described by an action (many are not). 2. and 4. must be assumed together.

Condition 1. is a consequence of the fact that since CFTs are scale invariant, they are fixed points of the renormalization group flow. Conversely, fixed points are scale invariant, but it is an open question whether that implies conformal invariance. But scale invariant QFTs are typically conformal.

In any case, for scaling the physical (i.e., renormalized) fields, $\Delta$ must be the true dimension, including anomalous terms (if there are any).

The proper definition is that all scalar n-point functions (or equivalently, their generating functional, a kind of partition function) remain unchanged when each field (including the metric, if it is a field) transforms according to a representation of the conformal group and the volume element (if the metric is not a field) is transformed by the standard conformal factor.

All other statements you mention are proxies for this, equivalent under additional assumptions only.

The proper definition of an (unbroken) conformal field theory is that all scalar n-point functions (or equivalently, their generating functional, a kind of partition function) remain unchanged when each field (including the metric, if it is a field) transforms according to a representation of the conformal group and the volume element (if the metric is not a field) is transformed by the standard conformal factor.

All other statements you mention are proxies for this, equivalent under additional assumptions only. For example, since you asked about conformal invariance but stated things about scale invariance only, you always need to add the requirement of Poincare co/invariance of the respective fields. Apart from that, you need to assume a specific field contents, and for 3. also that the CFT is described by an action (many are not). 2. and 4. must be assumed together.

Condition 1. is a consequence of the fact that CFTs are fixed points of the renormalization group flow. Conversely, fixed points are scale invariant, but it is an open question whether that implies conformal invariance. But scale invariant QFTs are typically conformal.

In any case, for scaling the physical (i.e., renormalized) fields, $\Delta$ must be the true dimension, including anomalous terms (if there are any).