Sorry but no; 13.5v out of the alternator is not working right. Residual magnetism in the field core can cause the alternator put out a low but inadequate voltage with VR not working or not connected.
DO us all a favor.... seriously:
GO to the store and buy and Brand new 9v transistor battery and measure the voltage out of that with your HF meter. It shoudl read 9.6 or 9.7 v. We *and you need to know if your meter is reading right since the voltages you are getting are in an indeterminate range. Another problem with the reported data is that the 13v out of the battery after a charge is too high IF you removed the charger and let it 'rest' for an hour or two before measuring; the 'surface charge' created in the battery in the charging process has to dissapate before a valid voltage measurement can be made. So, either you procedure is not right or the meter is inaccurate. We need to know which. Honestly, doing anything else is wasting your time and ours; these tests have to be done right to get useful results 'specially when help is over the internet.