Naive definition of treewidth
Your parameter "naive treewidth" is known as tree-partition-width in the literature.
It is known that if a graph has constant treewidth and constant maximum degree, then it has constant tree-partition-width. See [Wood 2009].

Note that your example in the update actually has pathwidth 2.

[Wood 2009] David R. Wood. On tree-partition-width. European Journal of Combinatorics 30 (2009) 1245-1253. https://doi.org/10.1016/j.ejc.2008.11.010
(also on arXiv https://arxiv.org/abs/math/0602507)