Home prices are not included in the shelter category. Homeowners are considered to be paying an implicit rent as their cost of shelter, which is based on the market rate of a comparable rental.

Your source for rentals looks at the cost of new leases. The BLS methodology looks at the most recent rent of a sample of units, regardless of when the rent was set.

We would expect your source to report a higher increase, because its methodology is more responsive to short term changes. That does not show that the BLS is lying.