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When I use the clip or intersect tools in QGIS it splits the polygon clipper features that are on the border of the clipper, which produces polygons that are only part of the real polygons.
Is there a tool that allows me to specify that features that are on the border are either in or out of clipping result set?
Try Vector > Geoprocessing Tools > Union and eventually remove a part of the polygons manually.
Would it work to use the Grass v.select function and then export as a new layer? It has an option called "Operator to use" which allows you to change the way the two layers interact in the selection process, i.e. intersect, contain, overlap.
I had the same issue, I solved this way (QGIS 3.4).
In my case, I needed to overlay a line shape with a polygon shape, keeping the whole lines intersecting the polygons without clipping them.
Step 1) Vector > Geoprocessing Tools > Intersect
Step 2) Vector > Geoprocessing Tools > Union with the intersected line layer and original line layer
