Does the word "may" in the California Code of Civil Procedure apply to a motion to amend the facts of a claim against a defendant?

Although the Code of Civil Procedure statutes use the word "may," thus committing the matter to the trial court's discretion, the "court must permit an amendment to conform to proof offered by plaintiff where no prejudice results to defendant." (Trainor v. Maus (1954) 126 Cal.App.2d 295, 302, 271 P.2d 957, citing Eckart v. Brown (1939) 34 Cal.App.2d 182, 188-189, 93 P.2d 212.) We see no reason to come to a different conclusion because a plaintiff's request is instead to "direct the fact to be found according to the evidence" as provided

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