The agronomist you spoke with is correct. You likely have a soil texture that is higher in sand particles than other types of clay soils, which would account for your lower CEC levels. Your soil texture is determined by the percentage of sand, silt, and clay in your soils.

Attached is a copy of the 'soil texture triangle'. If you can't open the attachment, just google that phrase and you'll find many versions of this same triangle. You'll see that there are actually soil texture designations called 'sandy clay' or 'sandy clay loam'. 

Despite the sand content, these clay soil types are still very easy to compact, so the areas where you have poor drainage are almost certainly where the soil is very compacted.

Yes, you probably have a sandy clay or sandy clay loam. If you are wanting to grow plants in the heavily compacted soil that does not drain well on your site, then you are correct to try to relieve that compaction before planting. 

There are a few ways to do this, depending on the scale of the site. Please note these recommendations are for areas where you currently do not have plants growing, as you would not want to do this under/around existing trees/shrubs in a way that would damage their roots.

Here is some info on soil compaction from University of Minnesota:

Soil compaction | UMN Extension

You can turn over the soil and hand mix in compost if creating a new garden bed. This article discusses using a broad fork as you suggested.

Reducing tillage in your garden | UMN Extension

If you are planning to plant a large area, you could use a machine. Here is a great info about breaking up compacted soils and incorporating compost on a larger scale, but it can also be done on a smaller scale. Keep in mind that you don't want to just relieve compaction and amend the soil with compost for each planting hole...you want to relive compaction and amend with compost across the whole planting bed/area. This link below also gives specific info on the type of compost to use.

Scoop and Dump Method of Soil Remediation – Summary | Urban Horticulture Institute (cornell.edu)

Good luck with your soil work!