I dug holes for the pergola too wide

My guess is this is just normal groundwater, either a natural spring or a high water table at that spot. (Other possibilities would be a leak, rainwater, or proximity to a septic system field.)

The water table height may be higher or lower depending on the season and amount of rain / snow. The water table may well be higher than the lake surface... they do not need to be the same height. The water table may be very irregular, depending on the underground position of rock, different types of soil, sources of water, etc, so just because there was no water somewhere else on your property doesn't preclude the possibility.

(The presence of water for only a brief time after digging is interesting, and I can't easily explain it. It's possible that by disturbing the soil your dogs were compacting it a bit, forcing water out, much like squeezing a wet sponge. This can be a problem in large-scale soil compaction projects, like preparing a lot for a new foundation. I would be surprised if dogs could have the same effect, but maybe if you have soft soil with a high water content it's possible.)

Use the core drill you mentioned before, make sure it is deep enough, say 4" to get to the bottom of the iron stub. That is the standard mounting depth.

To keep the drill in place, drill a hole in a piece of 3/4" plywood big enough (1'X4'??)to lay on the stair you need to drill with the hole in the place that the old post is located. Stand on this while you use the drill to cut the hole. Get another body on it too if you feel you need to, but if you can maintain good balance you will be fine. drill it down about 1/2" and set aside the plywood and finish the depth of the hole. you can always chip a little of the concrete out, to go deeper, but you will find that a well placed tap with a hammer and chisel or steel pin to the side will release the plug that you drilled.

The plywood will stay in place very well with the pressure of you standing on it, it does not take a whole lot of pressure to keep it from "dancing".

If you can use a water fed core drill.